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Let a knot be parameterized by a vector function with , and let be a fixed unit vector in . Count the number of local minima of the projection function . Then the minimum such number over all directions and all of the given type is called the crookedness . Milnor (1950) showed that is the infimum of the total curvature of . For any tame knot in , where is the bridge index.

The prime link 05-0201, illustrated above, with braid word or and Jones polynomialThe Whitehead link has linking number 0. It was discovered by Whitehead in 1934 (Whitehead 1962, pp. 21-50) as a counterexample to a piece of an attempted proof of the Poincaré conjecture (Milnor).

A link is said to be splittable if a plane can be embedded in such that the plane separates one or more components of from other components of and the plane is disjoint from . Otherwise, is said to be nonsplittable.The numbers of nonsplittable links (either prime or composite) with , 1, ... crossings are 1, 0, 1, 1, 3, 4, 15, ... (OEIS A086826).

A prime link is a link that cannot be represented as a knot sum of other links. Doll and Hoste (1991) list polynomials for oriented links of nine or fewer crossings, and Rolfsen (1976) gives a table of links with small numbers of components and crossings.The following table summarizes the number of distinct prime -components links having specified crossing numbers. ThecomponentsOEISprime -component links with 1, 2, ... crossings1A0028630, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, ...2A0489520, 1, 0, 1, 1, 3, 8, 16, 61, 185, 638, ...3A0489530, 0, 0, 0, 0, 3, 1, 10, 21, 74, 329, ...4A0870710, 0, 0, 0, 0, 0, 0, 3, 1, 15, 39, ...50, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, ...totalA0867710, 1, 1, 2, 3, 9, 16, 50, 132, 442, 1559, ...The following table lists some named links. The notation and ordering follows that of Rolfsen (1976), where denotes the th -component link with crossing number .link numbernameunlinkHopf linkWhitehead linkBorromean ringsA listing of the first few simple..

A planar diagram depicting a link (or knot) as a sequence of segments with gaps representing undercrossings and solid lines overcrossings. In such a diagram, only two segments should ever cross at a single point. Link diagrams for the trefoil knot and figure eight knot are illustrated above.

There are several different definition of link.In knot theory, a link is one or more disjointly embedded circles in three-space. More informally, a link is an assembly of knots with mutual entanglements. Kuperberg (1994) has shown that a nontrivial knot or link in has four collinear points (Eppstein). Like knots, links can be decomposed into basic units known as prime links.The term "link" is also used primarily by physicists to refer to a graphedge.

The prime link 02-0201 which has Jonespolynomialand HOMFLY polynomialIt has braid word .

A knot equivalent to a polygon in , also called a tame knot. For a polygonal knot , there exists a plane such that the orthogonal projection on it satisfies the following conditions: 1. The image has no multiple points other than a finite number of double points. 2. The projections of the vertices of are not double points of . Such a projection is called a regular knot projection.

The Perko pair is the pair of knots and illustrated above. For many years, they were listed as separate knots in Little (1885) and all similar tables, including the pictorial enumeration of Rolfsen (1976, Appendix C). They were identified as identical by Perko (1974), who found that they are related to one another by the so-called Perko move (Perko 1974, Hoste et al. 1998). Although these knots are equivalent, their diagrams have different writhes (Hoste et al. 1998).

The granny knot is a composite knot of six crossings consisting of a knot sum of two left-handed trefoils with the same orientation (Rolfsen 1976, p. 220).The granny knot has the same Alexander polynomial as the square knot. It also has the same group as the square knot (Rolfsen 1976, p. 62). However, the two knots are distinct, and Fox (1952) showed that the knot complements of the square and granny knots are not homeomorphic (Rolfsen 1976, p. 62). In addition, the square knot is a slice knot, while the granny knot is not (Rolfsen 1976, p. 220).The knot group of the granny knot iswhich is isomorphic to that of the square knot (Rolfsen1976, p. 62)

Given a doubled knot with the unknot taken as the base knot , the companion knot of is called a twist knot with twists. As illustrated above, the following knots are twist knots (Rolfsen 1976, p. 112).knottrefoil knot1figure eight knot2stevedore's knotThe unknot and stevedore's knot are the only twist knots that are slice knots (Casson and Gordon 1975; Rolfsen 1976, p. 226).

An oriented knot is an oriented link of one component, or equivalently, it is a knot which has been given an orientation. Given an oriented knot , reversing the orientation of may give rise to an inequivalent knot.Giving knots orientations are important to many applications of knot theory. Most importantly, providing orientations for knots allows for defining the sum of oriented knots simply by taking the connected sum of the knots regarded as oriented manifolds. Attempting to define a similar sum operation on non-oriented knots turns out not to be well-defined.As another example, knot orientations are necessary for producing Seifert surfaces for knots via the Seifert algorithm, which quite explicitly uses the orientation.

A knot or link in is said to be fibered if there exists a fibration and if the fibration is well-behaved near (Rolfsen 1976, p. 323).Examples of fibered knots include the trefoil knotand figure eight knot.The knot sum of two fibered knots is fibered (Rolfsen1976, p. 326).

A knot which is not alternating. Unlike alternating knots, flype moves are not sufficient to pass between all minimal diagrams of a given nonalternating knot (Hoste et al. 1998). In fact, Thistlethwaite used 13 different moves in generating a list of 16-crossing alternating knots (Hoste et al. 1998), and still had duplicates out of a list of knots (Hoste et al. 1998).The numbers of nonalternating knots with , 2, ... crossings are 0, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, ... (OEIS A051763), the first few of which are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .

A knot is an -embeddable knot if it can be placed on a genus standard embedded surface without crossings, but cannot be placed on any standardly embedded surface of lower genus without crossings. Any knot is an -embeddable knot for some . The figure eight knot is a 2-embeddable knot. A knot with bridge number is an -embeddable knot where .

A -torus knot is obtained by looping a string through the hole of a torus times with revolutions before joining its ends, where and are relatively prime. A -torus knot is equivalent to a -torus knot. All torus knots are prime (Hoste et al. 1998, Burde and Zieschang 2002). Torus knots are all chiral, invertible, and have symmetry group (Schreier 1924, Hoste et al. 1998).Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are torus knots using the function KnotData[knot, "Torus"].The link crossing number of a -torus knot is(1)(Williams 1988, Murasugi and Przytycki 1989, Murasugi 1991, Hoste et al. 1998). The unknotting number of a -torus knot is(2)(Adams 1991).The numbers of torus knots with crossings are 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, ... (OEIS A051764). Torus knots with fewer than 11 crossings are summarized in the following table (Adams et al. 1991) and the first few are illustrated above.knotnametrefoil..

Given an original knot , the knots produced by mutations together with itself are called mutant knots. Mutant knots are often difficult to distinguish. For instance, mutants have the same HOMFLY polynomials and hyperbolic knot volume. Many but not all mutants also have the same knot genus.

Let be the knot above, and let the homomorphism taking a knot to its companion knot be faithful (i.e., taking the preferred longitude and meridian of the original torus to the referred longitude and latitude of the tubular neighborhood). Then is an untwisted doubled knot of . Otherwise, is a doubled knot with twists (Rolfsen 1976, p. 112).

A knot embedded in , where the three-dimensional space is represented as a direct product of a complex line with coordinate and a real line with coordinate , in such a way that the coordinate is a Morse function on .

Conway's knot is the prime knot on 11 crossings withbraid wordThe Jones polynomial of Conway's knot iswhich is the same as for the Kinoshita-Terasakaknot.

A composite knot is a knot that is not a prime knot. Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed (Hoste et al. 1998).Knots that make up a knot sum of a composite knot areknown as factor knots.Combining prime knots gives no new knots for knots of three to five crossing, but two additional composite knots (the granny knot and square knot) with six crossings. The granny knot is the knot sum of two trefoils with the same chirality (), while the square knot is the knot sum of two trefoils with opposite chiralities (). There is a single composite knot of seven crossings () and four composite knots of eight crossings (, , , and ). The numbers of composite knots having , 2, ... crossings are therefore 0, 0, 0, 0, 0, 2, 1, 4, ......

Let be a knot that is geometrically essential in a standard embedding of the solid torus in the three-sphere . Let be another knot and let be a tubular neighborhood of in . Finally, let be a homeomorphism and let . Then is a companion knot of the knot (Rolfsen 1976, p. 111) and is called the satellite knot of (Adams 1994, pp. 115-118).

The square knot, also called the reef knot, is a composite knot of six crossings consisting of a knot sum of a trefoil knot and its mirror image (Rolfsen 1976, p. 220).The square knot has the same Alexander polynomial as the granny knot. It also has the same group as the granny knot (Rolfsen 1976, p. 62). However, the two knots are distinct. Fox (1952) showed that the knot complements of the square and granny knots are not homeomorphic (Rolfsen 1976, p. 62). In addition, the square knot is a slice knot, while the granny knot is not (Rolfsen 1976, p. 220).The knot group of the square knot iswhich is isomorphic to that of the granny knot (Rolfsen1976, p. 62).

In general, it is possible to link two -dimensional hyperspheres in -dimensional space in an infinite number of inequivalent ways. In dimensions greater than in the piecewise linear category, it is true that these spheres are themselves unknotted. However, they may still form nontrivial links. In this way, they are something like higher dimensional analogs of two one-spheres in three dimensions. The following table gives the number of nontrivial ways that two -dimensional hyperspheres can be linked in dimensions.D of spheresD of spacedistinct linkings234023931489591021813102182104383191021833Two 10-dimensional hyperspheres link up in 12, 13, 14, 15, and 16 dimensions, unlink in 17 dimensions, then link up again in 18, 19, 20, and 21 dimensions. The proof of these results consists of an "easy part" (Zeeman 1962) and "hard part" (Ravenel 1986). The hard part is related to the calculation of the (stable and unstable)..

A knot in is a slice knot if it bounds a disk in which has a tubular neighborhood whose intersection with is a tubular neighborhood for .Every ribbon knot is a slice knot, and it is conjecturedthat every slice knot is a ribbon knot.The knot determinant of a slice knot is a square number (Rolfsen 1976, p. 224).Slice knots include the unknot (Rolfsen 1976, p. 226), square knot (Rolfsen 1976, p. 220), stevedore's knot , and (Rolfsen 1976, p. 225), illustrated above.Casson and Gordon (1975) showed that the unknot and stevedore's knot are the only twist knots that are slice knots (Rolfsen 1976, p. 226).

The least genus of any Seifert surface for a given knot. The unknot is the only knot with genus 0.Usually, one denotes by the genus of the knot . The knot genus has the pleasing additivity property that if and are oriented knots, thenwhere the sum on the left hand side denotes knot sum. This additivity implies immediately, by induction, that any oriented knot can be factored into a sum of prime knots. Indeed, by the additivity of knot genus, any knot of genus 1 is prime. Furthermore, given any knot of genus , either itself is prime, or can be written as a sum of knots of lesser genus, each of which can be decomposed into a sum of prime knots, by induction.A nonobvious fact is that the prime decomposition is also unique.

A knot diagram is a picture of a projection of a knot onto a plane. Usually, only double points are allowed (no more than two points are allowed to be superposed), and the double or crossing points must be "genuine crossings" which transverse in the plane. This means that double points must look like the above left diagram, and not the above right one. Also, it is usually demanded that a knot diagram contain the information if the crossings are overcrossings or undercrossings so that the original knot can be reconstructed.The knot diagram of the trefoil knot is illustratedabove.Knot polynomials can be computed from knot diagrams. Such polynomials often (but not always) allow the knots corresponding to given diagrams to be uniquely identified.Rolfsen (1976) gives a table of knot diagrams for knots up to 10 crossings and links up to four components and 9 crossings. Adams (1994) gives a smaller table of knots diagrams up to 9 crossings, two-component..

An -bridge knot is a knot with bridge number . The set of 2-bridge knots is identical to the set of rational knots. If is a 2-bridge knot, then the BLM/Ho polynomial and Jones polynomial satisfywhere (Kanenobu and Sumi 1993). Kanenobu and Sumi also give a table containing the number of distinct 2-bridge knots of crossings for to 22, both not counting and counting mirror images as distinct. 30040056789104585119118212176341133527041469313651513872774162752546117550411008181096521845192193143862204377687381218755217510422174933349525

Let be a knot inside a torus, and knot the torus in the shape of a second knot (called the companion knot) , with certain additional mild restrictions to avoid trivial cases. Then the new knot resulting from is called the satellite knot . All satellite knots are prime (Hoste et al. 1998). The illustration above illustrates a satellite knot of the trefoil knot, which is the form all satellite knots of 16 or fewer crossings take (Hoste et al. 1998). Satellites of the trefoil share the trefoil's chirality, and all have wrapping number 2.Any satellite knot having wrapping number must have at least 27 crossings, and any satellite of the figure eight knot must have at least 17 crossings (Hoste et al. 1998). The numbers of satellite knots with crossings are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ... (OEIS A051765), so the satellite knot of minimal crossing number occurs for 13 crossings. A knot can be checked in the Wolfram Language to see if it is a satellite knot..

Let be the space in which a knot sits. Then the space "around" the knot, i.e., everything but the knot itself, is denoted and is called the knot complement of (Adams 1994, p. 84).If a knot complement is hyperbolic (in the sense that it admits a complete Riemannian metric of constant Gaussian curvature ), then this metric is unique (Prasad 1973, Hoste et al. 1998).

In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a "braid" rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.A knot can be generalized to a link, which is simply a knottedcollection of one or more closed strands.The study of knots and their properties is known as knot theory. Knot theory was given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations (Thompson 1867). P. G. Tait then..

If the knot is the boundary of a singular disk which has the property that each self-intersecting component is an arc for which consists of two arcs in , one of which is interior, then is said to be a ribbon knot.Every ribbon knot is a slice knot, and it is conjecturedthat every slice knot is a ribbon knot.Knot , illustrated above, is a ribbon knot (Rolfsen 1976, p. 225).

Let be a group, then there exists a piecewise linear knot in for with iff satisfies 1. is finitely presentable, 2. The Abelianization of is infinite cyclic, 3. The normal closure of some single element is all of , 4. ; the second homology of the group is trivial.

A knot is called prime if, for any decomposition as a connected sum, one of the factors is unknotted (Livingston 1993, pp. 5 and 78). A knot which is not prime is called a composite knot. It is often possible to combine two prime knots to create two different composite knots, depending on the orientation of the two. Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of prime knots.In general, it is nontrivial to determine if a given knot is prime or composite (Hoste et al. 1998). However, in the case of alternating knots, Menasco (1984) showed that a reduced alternating diagram represents a prime knot iff the diagram is itself prime ("an alternating knot is prime iff it looks prime"; Hoste et al. 1998).There is no known formula for giving the number of distinct prime knots as a function of the number of crossings. The numbers of distinct prime knots having..

A symmetry of a knot is a homeomorphism of which maps onto itself. More succinctly, a knot symmetry is a homeomorphism of the pair of spaces . Hoste et al. (1998) consider four types of symmetry based on whether the symmetry preserves or reverses orienting of and , 1. preserves , preserves (identity operation), 2. preserves , reverses , 3. reverses , preserves , 4. reverses , reverses . This then gives the five possible classes of symmetry summarized in the table below.classsymmetriesknot symmetries1chiral, noninvertible1, 3 amphichiral, noninvertible1, 4 amphichiral, noninvertible1, 2chiral, invertible1, 2, 3, 4 and amphichiral, invertibleIn the case of hyperbolic knots, the symmetry group must be finite and either cyclic or dihedral (Riley 1979, Kodama and Sakuma 1992, Hoste et al. 1998). The classification is slightly more complicated for nonhyperbolic knots. Furthermore, all knots with crossings are either amphichiral or invertible..

In the 1930s, Reidemeister first rigorously proved that knots exist which are distinct from the unknot. He did this by showing that all knot deformations can be reduced to a sequence of three types of "moves," called the (I) twist move, (II) poke move, and (III) slide move. These moves are most commonly called Reidemeister moves, although the term "equivalence moves" is sometimes also used (Aneziris 1999, p. 29).Reidemeister's theorem guarantees that moves I, II, and III correspond to ambient isotopy (moves II and III alone correspond to regular isotopy). He then defined the concept of colorability, which is invariant under Reidemeister moves.

A projection of a link is tricolorable if each of the strands in the projection can be colored in one of three different colors such that, at each crossing, all three colors come together or only one does and at least two different colors are used. The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and figure eight knot are not.If the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is.

A type I move (conjugation) takes for , where is a braid group.A type II move (stabilization) takes or for and , , and .

Two oriented knots (or links) can be summed by placing them side by side and joining them by straight bars so that orientation is preserved in the sum. The knot sum is also known as composition (Adams 1994) or connected sum (Rolfsen 1976, p. 40).This operation is denoted #, so the knot sum of knots and is writtenThe figure above illustrated the knot sum of two trefoil knots having the same handedness.The knot sum is in general not a well-defined operation, but depends on the choice of balls where the connection is made and perhaps also on the choice of the attaching homeomorphism. The square knot and granny knot illustrate this ambiguity (Rolfsen 1976, pp. 40-41).Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed. Knots that are the sums of prime knots are known as composite..

The span of an unoriented link diagram (also called the link spread) is the difference between the highest and lowest degrees of its bracket polynomial. The span is a topological invariant of a knot. If a knot has a reduced alternating projection of crossings, then the span of is .

The Conway polynomial , sometimes known as the Conway-Alexander polynomial, is a modified version of the Alexander polynomial that was formulated by J. H. Conway (Livingston 1993, pp. 207-215). It is a reparametrization of the Alexander polynomial given byThe skein relationship convention used by forthe Conway polynomial is(Doll and Hoste 1991).Examples of Alexander and Conway polynomials for common knots are given in the following tableknot trefoil knotfigure eight knotSolomon's seal knotstevedore's knotMiller Institute knot

A knot property, also called the twist number, defined as the sum of crossings of a link ,(1)where defined to be if the overpass slants from top left to bottom right or bottom left to top right and is the set of crossings of an oriented link.The writhe of a minimal knot diagram is not a knot invariant, as exemplified by the Perko pair, which have differing writhes (Hoste et al. 1998). This is because while the writhe is invariant under Reidemeister moves II and III, it may increase or decrease by one for a Reidemeister move of type I (Adams 1994, p. 153).Thistlethwaite (1988) proved that if the writhe of a reduced alternating projection of a knot is not 0, then the knot is not amphichiral (Adams 1994).A formula for the writhe is given by(2)where is parameterized by for along the length of the knot by parameter , and the frame associated with is(3)where is a small parameter, is a unit vector field normal to the curve at , and the vector field is given by(4)(Kaul..

Vassiliev invariants, discovered around 1989, provided a radically new way of looking at knots. The notion of finite type (a.k.a. Vassiliev) knot invariants was independently invented by V. Vassiliev and M. Goussarov around 1989. Vassiliev's approach is based on the study of discriminants in the (infinite-dimensional) spaces of smooth maps from one manifold into another. By definition, the discriminant consists of all maps with singularities.For example, consider the space of all smooth maps from the circle into three-space . If is an embedding (i.e., has no singular points), then it represents a knot. The complement of the set of all knots is the discriminant . It consists of all smooth maps from into that have singularities, either local, where , or nonlocal, where is not injective. Two knots are equivalent iff they can be joined by a path in the space that does not intersect the discriminant. Therefore, knot types are in one-to-one..

Kontsevich's integral is a far-reaching generalization of the Gauss integral for the linking number, and provides a tool to construct the universal Vassiliev invariant of a knot. In fact, any Vassiliev knot invariant can be derived from it.To construct the Kontsevich integral, represent the three-dimensional space as a direct product of a complex line with coordinate and a real line with coordinate . The integral is defined for Morse knots, i.e., knots embedded in in such a way that the coordinate is a Morse function on , and its values belong to the graded completion of the algebra of chord diagrams .The Kontsevich integral of the knot is defined as(1)where the ingredients of this formula have the following meanings. The real numbers and are the minimum and the maximum of the function on .The integration domain is the -dimensional simplex divided by the critical values into a certain number of connected components. For example, for the embedding..

The smallest number of times a knot must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techniques, but it is in general difficult to determine exact values. Many unknotting numbers can be determined from a knot's knot signature. A knot with unknotting number 1 is a prime knot (Scharlemann 1985). It is not always true that the unknotting number is achieved in a projection with the minimal number of crossings.The following table is from Kirby (1997, pp. 88-89), with the values for 10-139 and 10-152 taken from Kawamura (1998). In the following table, Kirby's (1997, p. 88) value has been corrected to reflect the fact that is only currently known to be 1 or 2 (Kawauchi 1996, p. 271). The value has been computed by Stoimenow (2002). The unknotting numbers for 10-154 and 10-161 can be found using the slice-Bennequin inequality (Stoimenow 1998).Knots for which the unknotting number..

The signature of a knot can be defined using the skein relationship(1)(2)and(3)where is the Conway polynomial and is an odd number.Many unknotting numbers can be determined usinga knot's signature.Knot signatures are implemented in the Wolfram Language as KnotData[knot, "Signature"]. The following table summarizes knot signatures for knots on 10 of fewer crossings.

The least number of unknotted arcs lying above the plane in any projection. The knot 05-002 has bridge number 2. Such knots are called 2-bridge knots. There is a one-to-one correspondence between 2-bridge knots and rational knots. The knot 08-010 is a 3-bridge knot. A knot with bridge number is an -embeddable knot where .

The twist of a ribbon measures how much it twists around its axis and is defined as the integral of the incremental twist around the ribbon. A formula for the twist is given by(1)where is parameterized by for along the length of the knot by parameter , and the frame associated with is(2)where is a small parameter and is a unit vector field normal to the curve at (Kaul 1999).Letting Lk be the linking number of the two components of a ribbon, Tw be the twist, and Wr be the writhe, then the calugareanu theorem states that(3)(Adams 1994, p. 187).

A knot invariant in the form of a polynomial such as the Alexander polynomial, BLM/Ho Polynomial, bracket polynomial, Conway polynomial, HOMFLY polynomial, Jones polynomial, Kauffman polynomial F, Kauffman polynomial X, and Vassiliev invariant.

A knot invariant is a function from the set of all knots to any other set such that the function does not change as the knot is changed (up to isotopy). In other words, a knot invariant always assigns the same value to equivalent knots (although different knots may have the same knot invariant). Standard knot invariants include the fundamental group of the knot complement, numerical knot invariants (such as Vassiliev invariants), polynomial invariants (knot polynomials such as the Alexander polynomial, Jones polynomial, Kauffman polynomial F, and Kauffman polynomial X), and torsion invariants (such as the torsion number).

The bracket polynomial is one-variable knot polynomial related to the Jones polynomial. The bracket polynomial, however, is not a topological invariant, since it is changed by type I Reidemeister moves. However, the polynomial span of the bracket polynomial is a knot invariant, as is a normalized form involving the writhe. The bracket polynomial is occasionally given the grandiose name regular isotopy invariant. It is defined by(1)where and are the "splitting variables," runs through all "states" of obtained by splitting the link, is the product of "splitting labels" corresponding to , and(2)where is the number of loops in .Letting(3)(4)gives a knot polynomial which is invariant under regular isotopy, and normalizing gives the Kauffman polynomial X which is invariant under ambient isotopy as well. The bracket polynomial of the unknot is 1. The bracket polynomial of the mirror image is the same as for..

A 1-variable unoriented knot polynomial . It satisfies(1)and the skein relationship(2)It also satisfies(3)where is the knot sum and(4)where is the mirror image of . The BLM/Ho polynomials of mutant knots are also identical. Brandt et al. (1986) give a number of interesting properties. For any link with components, is divisible by . If has components, then the lowest power of in is , and(5)where is the HOMFLY polynomial. Also, the degree of is less than the link crossing number of . If is a 2-bridge knot, then(6)where (Kanenobu and Sumi 1993).The polynomial was subsequently extended to the 2-variableKauffman polynomial F, which satisfies(7)Brandt et al. (1986) give a listing of polynomials for knots up to 8 crossings and links up to 6 crossings.

The Kauffman -polynomial, also called the normalized bracket polynomial, is a 1-variable knot polynomial denoted (Adams 1994, p. 153), (Kauffman 1991, p. 33), or (Livingston 1993, p. 219), and defined for a link by(1)where is the bracket polynomial and is the writhe of (Kauffman 1991, p. 33; Adams 1994, p. 153). It is implemented in the Wolfram Language as KnotData[knot, "BracketPolynomial"].This polynomial is invariant under ambientisotopy, and relates mirror images by(2)It is identical to the Jones polynomial with the change of variable(3)and related to the two-variable Kauffman polynomialF by(4)The Kaufman -polynomial of the trefoil knot is therefore(5)(Kaufmann 1991, p. 35). The following table summarizes the polynomials for named knots.knotKaufman -polynomialfigure eight knotMiller Institute knotPerko pairSolomon's seal knotstevedore's knottrefoil knotunknot1..

The arf invariant is a link invariant that always has the value 0 or 1. A knot has Arf invariant 0 if the knot is "pass equivalent" to the unknot and 1 if it is pass equivalent to the trefoil knot.Arf invariants are implemented in the Wolfram Language as KnotData[knot, "ArfInvariant"].The numbers of prime knots on , 2, ... crossings having Arf invariants 0 and 1 are summarized in the table below.OEIScounts of prime knots with , 2, ... crossings0A1314330, 0, 0, 0, 1, 1, 3, 10, 25, 82, ...1A1314340, 0, 1, 1, 1, 2, 4, 11, 24, 83, ...If , , and are projections which are identical outside the region of the crossing diagram, and and are knots while is a 2-component link with a nonintersecting crossing diagram where the two left and right strands belong to the different links, then(1)where is the linking number of and .The Arf invariant can be determined from the Alexander polynomial or Jones polynomial for a knot. For the Alexander polynomial..

Let the stick number of a knot be the least number of straight sticks needed to make a knot . The smallest stick number of any knot is , where is the trefoil knot. If and are knots, thenFor a nontrivial knot , let be the link crossing number (i.e., the least number of crossings in any projection of ). ThenStick numbers are implemented in the Wolfram Language as KnotData[knot, "StickNumber"].The following table gives the stick number for knots on 10 or fewer crossings.39111211121111106910121211121010791112121211101088101113111110108891211121210108910111112121110810101111121010108111011121311111091110111212101011910912111212101091010111213111091111111212101191010121012101191010111111101091010121213121010109121211111010119121112111110109121211101110101012121410101010911121111101011101211111010101091112111210101091212111111101110111111101110109121111101010101112111110101011111112111011101012111111111110111112131110111010111212121010..

A semi-oriented 2-variable knot polynomial definedby(1)where is an oriented link diagram, is the writhe of , is the unoriented diagram corresponding to , and is the bracket polynomial. It was developed by Kauffman by extending the BLM/Ho polynomial to two variables, and satisfies(2)The Kauffman polynomial is a generalization of the Jones polynomial since it satisfies(3)but its relationship to the HOMFLY polynomial is not well understood. In general, it has more terms than the HOMFLY polynomial, and is therefore more powerful for discriminating knots. It is a semi-oriented polynomial because changing the orientation only changes by a power of . In particular, suppose is obtained from by reversing the orientation of component , then(4)where is the linking number of with (Lickorish and Millett 1988). is unchanged by mutation.(5)(6)M. B. Thistlethwaite has tabulated the Kauffman 2-variable polynomialfor knots up to 13..

The algebraic unknotting number of a knot in is defined as the algebraic unknotting number of the -equivalence class of a Seifert matrix of . The algebraic unknotting number of an element in an -equivalent class is defined as the minimum number of algebraic unknotting operations necessary to transform the element to the -equivalence class of the zero matrix (Saeki 1999).

Solomon's seal knot is the prime (5,2)-torus knot with braid word . It is also known as the cinquefoil knot (a name derived from certain herbs and shrubs of the rose family which have five-lobed leaves and five-petaled flowers) or the double overhand knot. It has Arf invariant 1 and is not amphichiral, although it is invertible.The knot group of Solomon's seal knot is(1)(Livingston 1993, p. 127).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the Solomon's seal knot are(2)(3)(4)(5)(6)(7)Surprisingly, the knot 10-132 shares the same Alexander polynomial and Jones polynomial with the Solomon's seal knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial with it.

An invertible knot is a knot that can be deformed via an ambient isotopy into itself but with the orientation reversed. A knot that is not invertible is said to be noninvertible.Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are invertible using the command KnotData[knot, "Invertible"].Fox (1962, Problem 10, p. 169) pointed out several knots belonging to the standard table that seemed to be noninvertible. However, no noninvertible knots were proven to exist until Trotter (1964) discovered an infinite family, the smallest of which had 15 crossings.Three prime knots on 9 or fewer crossings are noninvertible: , , and (Cromwell 2004, pp. 297-299). Some noninvertible knots can be obtained in the Wolfram Language as KnotData["Noninvertible"]. The simplest noninvertible knot is (illustrated above) was first postulated to be noninvertible by Fox (1962; Whitten 1972).The..

P. G. Tait undertook a study of knots in response to Kelvin's conjecture that the atoms were composed of knotted vortex tubes of ether (Thomson 1869). He categorized knots in terms of the number of crossings in a plane projection. He also made some conjectures which remained unproven until the discovery of Jones polynomials: 1. Reduced alternating diagrams have minimal linkcrossing number, 2. Any two reduced alternating diagrams of a given knot have equal writhe,3. The flyping conjecture, which states that the number of crossings is the same for any reduced diagram of an alternating knot. Conjectures (1) and (2) were proved by Kauffman (1987), Murasugi (1987ab), and Thistlethwaite (1987, 1988) using properties of the Jones polynomial or Kauffman polynomial F (Hoste et al. 1998). Conjecture (3) was proved true by Menasco and Thistlethwaite (1991, 1993) using properties of the Jones polynomial (Hoste et al. 1998)...

A hyperbolic knot is a knot that has a complement that can be given a metric of constant curvature . All hyperbolic knots are prime knots (Hoste et al. 1998).A prime knot on 10 or fewer crossings can be tested in the Wolfram Language to see if it is hyperbolic using KnotData[knot, "Hyperbolic"].Of the prime knots with 16 or fewer crossings, all but 32 are hyperbolic. Of these 32, 12 are torus knots and the remaining 20 are satellites of the trefoil knot (Hoste et al. 1998). The nonhyperbolic knots with nine or fewer crossings are all torus knots, including (the -torus knot), , , (the -torus knot), and , the first few of which are illustrated above.The following table gives the number of nonhyperbolic and hyperbolic knots of crossing starting with .typeOEIScountstorusA0517641, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, ...satelliteA0517650, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ...nonhyperbolicA0524071, 0, 1, 0, 1, 1, 1, 1, 1, 0, 3, 3, 8, 11, ...hyperbolicA0524080,..

The set of fixed points which do not move as a knot is transformed into itself is not a knot. The conjecture was proved in 1978 (Morgan and Bass 1984). According to Morgan and Bass (1984), the Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it.The generalized Smith conjecture considers to be a piecewise linear -dimensional hypersphere in , and the -fold cyclic covering of branched along , and asks if is unknotted if is an (Hartley 1983). This conjecture is true for , and false for , with counterexamples in the latter case provided by Giffen (1966), Gordon (1974), and Sumners (1975).

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The th homotopy group of a topological space is the set of homotopy classes of maps from the n-sphere to , with a group structure, and is denoted . The fundamental group is , and, as in the case of , the maps must pass through a basepoint . For , the homotopy group is an Abelian group.The group operations are not as simple as those for the fundamental group. Consider two maps and , which pass through . The product is given by mapping the equator to the basepoint . Then the northern hemisphere is mapped to the sphere by collapsing the equator to a point, and then it is mapped to by . The southern hemisphere is similarly mapped to by . The diagram above shows the product of two spheres.The identity element is represented by the constant map . The choice of direction of a loop in the fundamental group corresponds to a manifold orientation of in a homotopy..

Let and be simplicial complexes, and let be a map. Suppose that whenever the vertices , ..., of span a simplex of , the points , ..., are vertices of a simplex of . Then can be extended to a continuous map such thatimpliesThe map is then called the linear simplicial map induced by the vertex map (Munkres 1993, p. 12).

Let be a bijective correspondence such that the vertices , ..., of span a simplex of iff , ..., span a simplex of . Then the induced simplicial map is a homeomorphism, and the map is called a simplicial homeomorphism (Munkres 1993, p. 13).

Two topological spaces and are homotopy equivalent if there exist continuous maps and , such that the composition is homotopic to the identity on , and such that is homotopic to . Each of the maps and is called a homotopy equivalence, and is said to be a homotopy inverse to (and vice versa).One should think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another.Certainly any homeomorphism is a homotopy equivalence, with homotopy inverse , but the converse does not necessarily hold.Some spaces, such as any ball , can be deformed continuously into a point. A space with this property is said to be contractible, the precise definition being that is homotopy equivalent to a point. It is a fact that a space is contractible, if and only if the identity map is null-homotopic, i.e., homotopic to a constant map...

Let and be CW-complexes and let (respectively ) denote the -skeleton of (respectively ). Then a continuous map is said to be cellular if it takes -skeletons to -skeletons for all , i.e, iffor all nonnegative integers .The contents of the cellular approximation theorem is that, in a certain sense, all maps between CW-complexes can be taken to be cellular.

Given two topological spaces and , place an equivalence relationship on the continuous maps using homotopies, and write if is homotopic to . Roughly speaking, two maps are homotopic if one can be deformed into the other. This equivalence relation is transitive because these homotopy deformations can be composed (i.e., one can follow the other).A simple example is the case of continuous maps from one circle to another circle. Consider the number of ways an infinitely stretchable string can be tied around a tree trunk. The string forms the first circle, and the tree trunk's surface forms the second circle. For any integer , the string can be wrapped around the tree times, for positive clockwise, and negative counterclockwise. Each integer corresponds to a homotopy class of maps from to .After the string is wrapped around the tree times, it could be deformed a little bit to get another continuous map, but it would still be in the same homotopy class,..

Let and be CW-complexes, and let be a continuous map. Then the cellular approximation theorem states that any such is homotopic to a cellular map. In fact, if the map is already cellular on a CW-subcomplex of , then the homotopy can be taken to be stationary on .A famous application of the theorem is the calculation of some homotopy groups of -spheres . Indeed, let and bestow on both and their usual CW-structure, with one -cell, and one -cell, respectively one -cell. If is a continuous, base-point preserving map, then by cellular approximation, it is homotopic to a cellular map . This map must map the -skeleton of into the -skeleton of , but the -skeleton of is itself, while the -skeleton of is the zero-cell, i.e., a point. This is because of the condition . Thus is a constant map, whence ...

The set , where is a closed star and is a star, is called the link of in a simplicial complex and is denoted (Munkres 1993, p. 11).

There are two types of bordism groups: bordism groups, also called cobordism groups or cobordism rings, and there are singular bordism groups. The bordism groups give a framework for getting a grip on the question, "When is a compact boundaryless manifold the boundary of another manifold?" The answer is, precisely when all its Stiefel-Whitney numbers are zero. Singular bordism groups give insight into Steenrod's realization problem: "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too.The machinery of the bordism group winds up being important for homotopytheory as well.

A simplicial complex is a space with a triangulation. Formally, a simplicial complex in is a collection of simplices in such that 1. Every face of a simplex of is in , and 2. The intersection of any two simplices of is a face of each of them (Munkres 1993, p. 7).Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes. When only simplicial complexes and simplicial subcomplexes are considered, defining homology is particularly easy (and, in fact, combinatorial because of its finite/counting nature). This kind of homology is called simplicial homology.

A continuous transformation from one function to another. A homotopy between two functions and from a space to a space is a continuous map from such that and , where denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping space from the first function to the second.Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. The concept of homotopy was first formulated by Poincaré around 1900 (Collins 2004).

If is a simple closed curve in , the closure of one of the components of is homeomorphic with the unit 2-ball. This theorem may be proved using the Riemann mapping theorem, but the easiest proof is via Morse theory.The generalization to dimensions is called Mazur's theorem. It follows from the Schönflies theorem that any two knots of in or are equivalent.

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps and are homotopic if there is a continuous mapsuch that and .Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane . The puncture can be thought of as an obstacle.However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces and are homotopy equivalent if there are maps and such that the composition is homotopic to the identity map of and is homotopic to the identity map of . For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible..

Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that remain invariant under both-directions continuous one-to-one (homeomorphic) transformations. The discipline of algebraic topology is popularly known as "rubber-sheet geometry" and can also be viewed as the study of disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of hole structures, and it gets the prefix "algebraic" since many hole structures are represented best by algebraic objects like groups and rings.Algebraic topology originated with combinatorial topology, but went beyond it probably for the first time in the 1930s when Čech cohomology was developed.A technical way of saying this is that algebraic topology is concerned with functors from the topological..

A subspace of is called a retract of if there is a continuous map (called a retraction) such that for all and all , 1. , and 2. . Equivalently, a subspace of is called a retract of if there is a continuous map (called a retraction) such that for all ,

Let denote a chain complex, a portion of which is shown below:Let denotes the th homology group. Then two homology cycles are said to be homologous, if their difference is a boundary, i.e., if .

Let be a continuum (i.e., a compact connected metric space). Then is hereditarily unicoherent provided that every subcontinuum of is unicoherent.Any hereditarily unicoherent continuum is a unicoherent space, but there are unicoherent continua that are not hereditarily unicoherent. For example, the unit interval is hereditarily unicoherent, but a ray winding down on a circle is not hereditarily unicoherent, even though it is unicoherent. (This is due to the fact that a circle is not unicoherent.)

Let be a class of topological spaces that is closed under homeomorphism, and let be a topological space. If and for every such that , is a retract of , then is an absolute retract for the class .These notions can be generalized to category theory, and because there are category-theoretic versions, there are also other more specific versions, as in universal algebra and modern algebra.

An invariant defined using the angles of a three-dimensional polyhedron. It remains constant under solid dissection and reassembly. Solids with the same volume can have different Dehn invariants.Two polyhedra can be dissected into each other only if they have the same volume and the same Dehn invariant. In 1902, Dehn showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems, and Sydler (1965) showed that two polyhedra with the same Dehn invariants are interdissectable.

A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function in its entire domain (i.e., ) does not have compact support, while any bump function does have compact support.

A topological space is semilocally simply connected (also called semilocally 1-connected) if every point has a neighborhood such that any loop with basepoint is homotopic to the trivial loop. The prefix semi- refers to the fact that the homotopy which takes to the trivial loop can leave and travel to other parts of .The property of semilocal simple connectedness is important because it is a necessary and sufficient condition for a connected, locally pathwise-connected space to have a universal cover.

There are several equivalent definitions of a closed set. Let be a subset of a metric space. A set is closed if 1. The complement of is an open set, 2. is its own set closure, 3. Sequences/nets/filters in that converge do so within , 4. Every point outside has a neighborhood disjoint from . The point-set topological definition of a closed set is a set which contains all of its limit points. Therefore, a closed set is one for which, whatever point is picked outside of , can always be isolated in some open set which doesn't touch .The most commonly encountered closed sets are the closed interval, closed path, closed disk, interior of a closed path together with the path itself, and closed ball. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points (and is nowhere dense, so it has Lebesgue measure 0).It is possible for a set to be neither open nor closed, e.g., the half-closed interval ...

An -dimensional closed disk of radius is the collection of points of distance from a fixed point in -dimensional Euclidean space. Krantz (1999, p. 3) uses the symbol to denote the closed disk, and to denote the unit closed disk centered at the origin

The topology induced by a topological space on a subset . The open sets of are the intersections , where is an open set of .For example, in the relative topology of the interval induced by the Euclidean topology of the real line, the half-open interval is open since it coincides with . This example shows that an open set of the relative topology of need not be open in the topology of .

A mathematical structure is said to be closed under an operation if, whenever and are both elements of , then so is .A mathematical object taken together with its boundary is also called closed. For example, while the interior of a sphere is an open ball, the interior together with the sphere itself is a closed ball.

A Reinhardt domain with center is a domain in such that whenever contains , the domain also contains the closed polydisk.

An outer measure on is Borel regular if, for each set , there exists a Borel set such that . The -dimensional Hausdorff outer measure is regular on .

The generalized diameter is the greatest distance between any two points on the boundary of a closed figure. The diameter of a subset of a Euclidean space is therefore given bywhere denotes the supremum (Croft et al. 1991).For a solid object or set of points in Euclidean -space, the generalized diameter is equal to the generalized diameter of its convex hull. This means, for example, that the generalized diameter of a polygon or polyhedron can be found simply by finding the greatest distance between any two pairs of vertices (without needing to consider other boundary points).The generalized diameter is related to the geometricspan of a set of points.

Each point in the convex hull of a set in is in the convex combination of or fewer points of .

A Cartesian product of any finite or infinite set of copies of , equipped with the product topology derived from the discrete topology of . It is denoted . The name is due to the fact that for , this set is closely related to the Cantor set (which is formed by all numbers of the interval which admit an expansion in base 3 formed by 0s and 2s only), and this gives rise to a one-to-one correspondence between and the Cantor set, which is actually a homeomorphism. In the symbol denoting the Cantor discontinuum, can be replaced by 2 and by .

Let be an -manifold and let denote a partition of into disjoint pathwise-connected subsets. Then if is a foliation of , each is called a leaf and is not necessarily closed or compact.

A set in a metric space is bounded if it has a finite generalized diameter, i.e., there is an such that for all . A set in is bounded iff it is contained inside some ball of finite radius (Adams 1994).

Let be an -manifold and let denote a partition of into disjoint pathwise-connected subsets. Then is called a foliation of of codimension (with ) if there exists a cover of by open sets , each equipped with a homeomorphism or which throws each nonempty component of onto a parallel translation of the standard hyperplane in . Each is then called a foliation leaf and is not necessarily closed or compact (Rolfsen 1976, p. 284).

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space with more than one connected component.A subset of the Euclidean plane with more than one element can always be disconnected by cutting it through with a line (i.e., by taking out its intersection with a suitable straight line). In fact, it is certainly possible to find a line such that two points of lie on different sides of . If the Cartesian equation of is(1)for fixed real numbers , then the set is disconnected, since it is the union of the two nonempty open subsets(2)and(3)which are the sets of elements of lying on the two sides of .

A subset is said to be bicollared in if there exists an embedding such that when . The map or its image is then said to be the bicollar.

One of the Eilenberg-Steenrod axioms. Let be a single point space. unless , in which case where are some groups. The are called the coefficients of the homology .

A topology defined on a totally ordered set whose open sets are all the finite intersections of subsets of the form or , where .The order topology of the real line is the Euclidean topology. The order topology of is the discrete topology, since for all ,is an open set.

Let be a subset of a metric space. Then the set is open if every point in has a neighborhood lying in the set. An open set of radius and center is the set of all points such that , and is denoted . In one-space, the open set is an open interval. In two-space, the open set is a disk. In three-space, the open set is a ball.More generally, given a topology (consisting of a set and a collection of subsets ), a set is said to be open if it is in . Therefore, while it is not possible for a set to be both finite and open in the topology of the real line (a single point is a closed set), it is possible for a more general topological set to be both finite and open.The complement of an open set is a closed set. It is possible for a set to be neither open nor closed, e.g., the half-closed interval ...

A set in a first-countable space is dense in if , where is the set of limit points of . For example, the rational numbers are dense in the reals. In general, a subset of is dense if its set closure .A real number is said to be -dense iff, in the base- expansion of , every possible finite string of consecutive digits appears. If is -normal, then is also -dense. If, for some , is -dense, then is irrational. Finally, is -dense iff the sequence is dense (Bailey and Crandall 2001, 2003).

Let be a topological space, and let . Then the arc component of is

Suppose that are arcwise-connected and locally arcwise-connected topological spaces. Then is said to be a covering space of if is a surjective continuous map with every having an open neighborhood such that every connected component of is mapped homeomorphically onto by .

A weakened version of pointwise convergence hypothesis which states that, for a measure space, for all , where is a measurable subset of such that .

For any point on the boundary of an ordinary ball, find a neighborhood of in which the intersection with the ball's boundary cuts the neighborhood into two parts, each homeomorphic to a ball. A wild point is a point on the boundary that has no such neighborhood.

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The th Chern class is an obstruction to the existence of everywhere complex linearly independent vector fields on that vector bundle. The th Chern class is in the th cohomology group of the base space.

An -dimensional manifold is said to be a homotopy sphere, if it is homotopy equivalent to the -sphere . Thus no homotopy group can distinguish between and .The solution of the generalized Poincaré conjecture in the positive implies that any compact homotopy sphere is in fact homeomorphic to a sphere.

Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the existence of everywhere linearly independent vector fields on the vector bundle. The most common examples of characteristic classes are the Chern, Pontryagin, and Stiefel-Whitney classes.

The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements in a neighborhood set of . 1. There corresponds to each point at least one neighborhood , and each neighborhood contains the point . 2. If and are two neighborhoods of the same point , there must exist a neighborhood that is a subset of both. 3. If the point lies in , there must exist a neighborhood that is a subset of . 4. For two different points and , there are two corresponding neighborhoods and with no points in common.

One of the Eilenberg-Steenrod axioms which states that, if is homotopic to , then their induced maps and are the same.

The Lebesgue covering dimension is an important dimension and one of the first dimensions investigated. It is defined in terms of covering sets, and is therefore also called the covering dimension (as well as the topological dimension).A space has Lebesgue covering dimension if for every open cover of that space, there is an open cover that refines it such that the refinement has order at most . Consider how many elements of the cover contain a given point in a base space. If this has a maximum over all the points in the base space, then this maximum is called the order of the cover. If a space does not have Lebesgue covering dimension for any , it is said to be infinite dimensional.Results of this definition are: 1. Two homeomorphic spaces have the same dimension, 2. has dimension , 3. A topological space can be embedded as a closed subspace of a Euclidean space iff it is locally compact, T2, second countable, and is finite-dimensional (in the sense of the..

A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold.For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold.For any positive integer , a distinct nonorientable..

A pair , where is a manifold and is a symplectic form on . The phase space is a symplectic manifold. Near every point on a symplectic manifold, it is possible to find a set of local "Darboux coordinates" in which the symplectic form has the simple form(Sjamaar 1996), where is a wedge product.

A sutured manifold is a tool in geometric topology which was first introduced by David Gabai in order to study taut foliations on 3-manifolds. Roughly, a sutured manifold is a pair with a compact, oriented 3-manifold with boundary and with a set of simple closed curves in which are oriented and which divide into pieces and (Juhász 2010).Defined precisely in a seminal work by Gabai (1983), a sutured manifold is a compact oriented 3-manifold together with a set of pairwise disjoint annuli and tori such that each component of contains a homologically nontrivial oriented simple closed curve (called a suture) and such that is oriented. Using this construction, the collection of a sutured manifold effectively splits into disjoint pieces and with , respectively , defined to be the components of whose normal vectors point into, respectively point out of, . Gabai's definition also requires that orientations on be coherent with respect to the..

Cohomology is an invariant of a topological space, formally "dual" to homology, and so it detects "holes" in a space. Cohomology has more algebraic structure than homology, making it into a graded ring (with multiplication given by the so-called "cup product"), whereas homology is just a graded Abelian group invariant of a space.A generalized homology or cohomology theory must satisfy all of the Eilenberg-Steenrodaxioms with the exception of the dimension axiom.

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point forms a vector space called the tangent space at , and the collection of tangent spaces on a manifold forms a vector bundle called the tangent bundle.A tangent vector at a point on a manifold is a tangent vector at in a coordinate chart. A change in coordinates near causes an invertible linear map of the tangent vector's representations in the coordinates. This transformation is given by the Jacobian, which must be nonsingular in a change of coordinates. Hence the tangent vectors at are well-defined. A vector field is an assignment of a tangent vector for each point. The collection of tangent vectors forms the tangent bundle, and a vector field is a section of this bundle.Tangent vectors are used to do calculus on manifolds. Since manifolds are locally Euclidean, the usual notions of differentiation and integration..

A submersion is a smooth map whengiven that the differential, or Jacobian, is surjective at every in . The basic example of a submersion is the canonical submersion of onto when ,In fact, if is a submersion, then it is possible to find coordinates around in and coordinates around in such that is the canonical submersion written in these coordinates. For example, consider the submersion of onto the circle , given by .

An orientation on an -dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an orientation exists on , then is called orientable.Not all manifolds are orientable, as exemplified by the Möbius strip and the Klein bottle, illustrated above.However, an -dimensional submanifold of is orientable iff it has a unit normal vector field. The choice of unit determines the orientation of the submanifold. For example, the sphere is orientable.Some types of manifolds are always orientable. For instance, complex manifolds, including varieties, and also symplectic manifolds are orientable. Also, any unoriented manifold has a double cover which is oriented.A map between oriented manifolds of the same dimension is called orientation preserving if the volume form on pulls back to a positive volume form on . Equivalently, the differential maps an oriented..

Let be a non-wandering point of a diffeomorphism of a compact manifold. The closing lemma concerns if can be arbitrarily well approximated with derivatives of order for each by so that is a periodic point of .The closing lemma is the 10th of Smale's problemsand remains unsettled.

A polynomial which is not necessarily an invariant of a link. It is related to the dichroic polynomial. It is defined by the skein relationship(1)and satisfies(2)and(3)

The second knot polynomial discovered. Unlike the first-discovered Alexander polynomial, the Jones polynomial can sometimes distinguish handedness (as can its more powerful generalization, the HOMFLY polynomial). Jones polynomials are Laurent polynomials in assigned to an knot. The Jones polynomials are denoted for links, for knots, and normalized so that(1)For example, the right-hand and left-hand trefoil knotshave polynomials(2)(3)respectively.If a link has an odd number of components, then is a Laurent polynomial over the integers; if the number of components is even, is times a Laurent polynomial. The Jones polynomial of a knot sum satisfies(4)The skein relationship for under- and overcrossingsis(5)Combined with the link sum relationship, this allows Jones polynomials to be built up from simple knots and links to more complicated ones.Some interesting identities from Jones (1985) follow. For any link ,(6)where is the..

The Alexander polynomial is a knot invariant discovered in 1923 by J. W. Alexander (Alexander 1928). The Alexander polynomial remained the only known knot polynomial until the Jones polynomial was discovered in 1984. Unlike the Alexander polynomial, the more powerful Jones polynomial does, in most cases, distinguish handedness.In technical language, the Alexander polynomial arises from the homology of the infinitely cyclic cover of a knot complement. Any generator of a principal Alexander ideal is called an Alexander polynomial (Rolfsen 1976). Because the Alexander invariant of a tame knot in has a square presentation matrix, its Alexander ideal is principal and it has an Alexander polynomial denoted .Let be the matrix product of braid words of a knot, then(1)where is the Alexander polynomial and det is the determinant. The Alexander polynomial of a tame knot in satisfies(2)where is a Seifert matrix, det is the determinant,..

Two knots are pass equivalent if there exists a sequence of pass moves taking one to the other. Every knot is either pass equivalent to the unknot or trefoil knot. These two knots are not pass equivalent to each other, but the enantiomers of the trefoil knot are pass equivalent. A knot has Arf invariant 0 if the knot is pass equivalent to the unknot and 1 if it is pass equivalent to the trefoil knot.

An Alexander matrix is a presentation matrix for the Alexander invariant of a knot . If is a Seifert matrix for a tame knot in , then and are Alexander matrices for , where denotes the transpose.

A 2-variable oriented knot polynomial motivated by the Jones polynomial (Freyd et al. 1985). Its name is an acronym for the last names of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter (Freyd et al. 1985). Independent work related to the HOMFLY polynomial was also carried out by Prztycki and Traczyk (1987). HOMFLY polynomial is defined by the skein relationship(1)(Doll and Hoste 1991), where is sometimes written instead of (Kanenobu and Sumi 1993) or, with a slightly different relationship, as(2)(Kauffman 1991). It is also defined as in terms of skein relationship(3)(Lickorish and Millett 1988). It can be regarded as a nonhomogeneous polynomial in two variables or a homogeneous polynomial in three variables. In three variables the skein relationship is written(4)It is normalized so that . Also, for unlinked unknotted components,(5)This polynomial usually detects chirality but does not detect the distinct..

The Alexander invariant of a knot is the homology of the infinite cyclic cover of the complement of , considered as a module over , the ring of integral laurent polynomials. The Alexander invariant for a classical tame knot is finitely presentable, and only is significant.For any knot in whose complement has the homotopy type of a finite CW-complex, the Alexander invariant is finitely generated and therefore finitely presentable. Because the Alexander invariant of a tame knot in has a square presentation matrix, its Alexander ideal is principal and it has an Alexander polynomial denoted .

A link invariant defined for a two-component oriented link as the sum of crossings and crossing over all crossings between the two links divided by 2. For components and ,where is the set of crossings of with , and is the sign of the crossing. The linking number of a splittable two-component link is always 0.

Consider two closed oriented space curves and , where and are distinct circles, and are differentiable functions, and and are disjoint loci. Let be the linking number of the two curves, then the Gauss integral is

Given a Seifert form , choose a basis , ..., for as a -module so every element is uniquely expressible as(1)with integer. Then define the Seifert matrix as the integer matrix with entries(2)For example, the right-hand trefoil knot has Seifertmatrix(3)A Seifert matrix is not a knot invariant, but it can be used to distinguish between different Seifert surfaces for a given knot.

A simple way to describe a knot projection. The advantage of this notation is thatit enables a knot diagram to be drawn quickly.For an oriented alternating knot with crossings, begin at an arbitrary crossing and label it 1. Now follow the undergoing strand to the next crossing, and denote it 2. Continue around the knot following the same strand until each crossing has been numbered twice. Each crossing will have one even number and one odd number, with the numbers running from 1 to .Now write out the odd numbers 1, 3, ..., in a row, and underneath write the even crossing number corresponding to each number. The Dowker notation is this bottom row of numbers. When the sequence of even numbers can be broken into two permutations of consecutive sequences (such as ), the knot is composite and is not uniquely determined by the Dowker notation. Otherwise, the knot is prime and the notation uniquely defines a single knot (for amphichiral knots) or corresponds..

For a given knot in , choose a Seifert surface in for and a bicollar in . If is represented by a 1-cycle in , let denote the homology cycle carried by in the bicollar. Similarly, let denote . The function defined bywhere lk denotes the linking number, is called a Seifert form for .

A concise notation based on the concept of the tangle used by Conway (1967) to enumerate prime knots up to 11 crossings.An algebraic knot containing no negative signs in its Conway knot notation is an alternating knot.Conway's knot notation is implemented in the Wolfram Language as KnotData[knot, "ConwayNotation"]. Rolfsen (1976) gives a table that includes Conway's knot notation for prime knots on 10 or fewer crossings, as summarized in the table below.

Let and be disjoint bicollared knots in or and let denote the open region between them. Then the closure of is a closed annulus . Except for the case , the theorem was proved by Kirby (1969).

A region in a knot or link projection plane surrounded by a circle such that the knot or link crosses the circle exactly four times. Two tangles are equivalent if a sequence of Reidemeister moves can be used to transform one into the other while keeping the four string endpoints fixed and not allowing strings to pass outside the circle.The simplest tangles are the -tangle and 0-tangle, shown above. A tangle with left-handed twists is called an -tangle, and one with right-handed twists is called a -tangle. By placing tangles side by side, more complicated tangles can be built up such as (, 3, 2), etc. The link created by connecting the ends of the tangles is now described by the sequence of tangle symbols, known as Conway's knot notation. If tangles are multiplied by 0 and then added, the resulting tangle symbols are separated by commas. Additional symbols which are used are the period, colon, and asterisk.Amazingly enough, two tangles described in this..

A closed curve associated with a knot which is displaced along the normal by a small amount. For a knot parameterized as for along the length of the knot by parameter , the frame associated with iswhere is a small parameter, is a unit vector field normal to the curve at .

A relationship between knot polynomials for links in different orientations (denoted below as , , and ). J. H. Conway was the first to realize that the Alexander polynomial could be defined by a relationship of this type.

Gives a matrix representation of a braid group in terms of matrices. A always appears in the position.(1)(2)(3)Let be the matrix product of braid words, then(4)where is the Alexander polynomial and det is the determinant.

A braid is an intertwining of some number of strings attached to top and bottom "bars" such that each string never "turns back up." In other words, the path of each string in a braid could be traced out by a falling object if acted upon only by gravity and horizontal forces. A given braid may be assigned a symbol known as a braid word that uniquely identifies it (although equivalent braids may have more than one possible representations). In particular, an -braid can constructed by iteratively applying the () operator, which switches the lower endpoints of the th and th strings--keeping the upper endpoints fixed--with the th string brought above the th string. If the th string passes below the th string, it is denoted .An ordered combination of the and symbols constitutes a braid word. For example, is a braid word for the braid illustrated above, where the symbols can be read off the diagram left to right and then top to bottom.By Alexander's..

A braid index is the least number of strings needed to make a closed braid representation of a link. The braid index is equal to the least number of Seifert circles in any projection of a knot (Yamada 1987). Also, for a nonsplittable link with link crossing number and braid index ,(Ohyama 1993). Let be the largest and the smallest power of in the HOMFLY polynomial of an oriented link, and be the braid index. Then the morton-franks-williams inequality holds,(Franks and Williams 1987). The inequality is sharp for all prime knots up to 10 crossings with the exceptions of 09-042, 09-049, 10-132, 10-150, and 10-156.

A braid is an intertwining of some number of strings attached to top and bottom "bars" such that each string never "turns back up." In other words, the path of each string in a braid could be traced out by a falling object if acted upon only by gravity and horizontal forces. A given braid may be assigned a symbol known as a braid word that uniquely identifies it (although equivalent braids may have more than one possible representations). For example, is a braid word for the braid illustrated above.If is a knot andwhere is the Alexander polynomial of , then cannot be represented as a closed 3-braid. Also, ifthen cannot be represented as a closed 4-braid (Jones 1985).

An amphichiral knot is a knot that is capable of being continuously deformed into its own mirror image. More formally, a knot is amphichiral (also called achiral or amphicheiral) if there exists an orientation-reversing homeomorphism of mapping to itself (Hoste et al. 1998). (If the words "orientation-reversing" are omitted, all knots are equivalent to their mirror images.)Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are amphichiral using the command KnotData[knot, "Amphichiral"].There are 20 amphichiral knots having ten or fewer crossings, namely (the figure eight knot), , , , , , , , , , , , , , , , , , , and (Jones 1985), the first few of which are illustrated above.The following table gives the total number of prime amphichiral knots, number of amphichiral noninvertible prime knots, amphichiral noninvertible prime knots, and fully amphichiral invertible knots prime knots () with..

The unknot, also called the trivial knot (Rolfsen 1976, p. 51), is a closed loop that is not knotted. In the 1930s Reidemeister first proved that knots exist which are distinct from the unknot by inventing and making use of the so-called Reidemeister moves and coloring each part of a knot diagram with one of three colors.The unknot is implemented in the WolframLanguage as KnotData["Unknot"].The knot sum of two unknots is another unknot.The Jones polynomial of the unknot is definedto give the normalization(1)The unknot has Alexander polynomial and Conway polynomial (2)(3)Surprisingly, there are known examples of nontrivial knots with Alexander polynomial 1, although no such examples occur among the knots of 10 or fewer crossings. An example is the -pretzel knot (Adams 1994, p. 167). Rolfsen (1976, p. 167) gives four other such examples.Haken (1961) devised an algorithm to tell if a knot projection is the unknot...

The stevedore's knot is the 6-crossing prime knot . It is implemented in the Wolfram Language as KnotData["Stevedore"].It has braid word . It has Arf invariant 0 and is not amphichiral, although it is invertible. It is a slice knot (Rolfsen 1976, p. 225).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , and Jones polynomial of Stevedore's knot are(1)(2)(3)(4)(5)Surprisingly, the knot 09-046 shares the same Alexander polynomial with the stevedore's knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial or Jones polynomial with it.

The Miller Institute knot is the 6-crossing prime knot . It is alternating, chiral, and invertible. A knot diagram of its laevo form is illustrated above, which is implemented in the Wolfram Language as KnotData[6, 2].The knot is so-named because it appears on the logo of the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science at the University of California, Berkeley (although, as can be seen in the logo, the Miller Institute's knot actually has dextro chirality).The knot has braid word . It has Arf invariant 1 and is not amphichiral, although it is invertible.The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , and Jones polynomial of the Miller Institute knot are(1)(2)(3)(4)(5)No knots on 10 or fewer crossings share the same Alexander polynomial, BLM/Ho polynomial, or Jones polynomial with the Miller Institute knot...

An alternating knot is a knot which possesses a knot diagram in which crossings alternate between under- and overpasses. Not all knot diagrams of alternating knots need be alternating diagrams.The trefoil knot and figure eight knot are alternating knots, as are all prime knots with seven or fewer crossings. A knot can be checked in the Wolfram Language to see if it is alternating using KnotData[knot, "Alternating"].The number of prime alternating and nonalternating knots of crossings are summarized in the following table.typeOEIScountsalternatingA0028640, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878, 19536, 85263, 379799, ...nonalternatingA0517630, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, 5110, 27436, 168030, 1008906, ...The 3 nonalternating knots of eight crossings are , , and , illustrated above (Wells 1991).One of Tait's knot conjectures states that the number of crossings is the same for any diagram of a reduced alternating..

Call a projection of a link an almost alternating projection if one crossing change in the projection makes it an alternating projection. Then an almost alternating link is a link with an almost alternating projection, but no alternating projection. Every alternating knot has an almost alternating projection. A prime knot which is almost alternating is either a torus knot or a hyperbolic knot. Therefore, no satellite knot is an almost alternating knot.All nonalternating 9-crossing prime knots are almost alternating. Of the 393 nonalternating knots and links with 11 or fewer crossings, all but five are known to be almost alternating (and 3 of these have 11 crossings). The fate of the remaining five is not known. The -, -, and -torus knots are almost alternating (Adams 1994, p. 142).

A class of knots containing the class of alternating knots. Let be the link crossing number. Then for knot sum which is an adequate knot,This relationship is postulated to hold true for all knots.

The trefoil knot , also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word . The trefoil and its mirror image are not equivalent, as first proved by Dehn (1914). In other words, the trefoil knot is not amphichiral. It is, however, invertible, and has Arf invariant 1.Its laevo form is implemented in the WolframLanguage, as illustrated above, as KnotData["Trefoil"].M. C. Escher's woodcut "Knots" (Bool et al. 1982, pp. 128 and 325; Forty 2003, Plate 71) depicts three trefoil knots composed of differing types of strands. A preliminary study (Bool et al. 1982, p. 123) depicts another trefoil.The animation above shows a series of gears arranged along a Möbiusstrip trefoil knot (M. Trott).The bracket polynomial can be computed as follows.(1)(2)Plugging in(3)(4)gives(5)The corresponding Kauffman polynomial..

The Borromean rings, also called the Borromean links (Livingston 1993, p. 10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).The Borromean rings are a prime link. They have link symbol 06-0302, braid word , and are also the simplest Brunnian link.It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings...

A braid with strands and components with positive crossings and negative crossings satisfieswhere is the unknotting number. While the second part of the inequality was already known to be true (Boileau and Weber, 1983, 1984) at the time the conjecture was proposed, the proof of the entire conjecture was completed using results of Kronheimer and Mrowka on Milnor's conjecture (and, independently, using the slice-Bennequin inequality).

Given a knot diagram, it is possible to construct a collection of variables and equations, and given such a collection, a group naturally arises that is known as the group of the knot. While the group itself depends on the choices made in the construction, any two groups that arise in this way are isomorphic (Livingston 1993, p. 103).For example, the knot group of the trefoil knot is(1)or equivalently(2)(Rolfsen 1976, pp. 52 and 61), while that of Solomon'sseal knot is(3)(Livingston 1993, p. 127).The group of a knot is not a complete knot invariant (Rolfsen 1976, p. 62). Furthermore, it is often quite difficult to prove that two knot group presentations represent nonisomorphic groups (Rolfsen 1976, p. 63).

A set is said to be bounded from below if it has a lowerbound.Consider the real numbers with their usual order. Then for any set , the infimum exists (in ) if and only if is bounded from below and nonempty.

For a point , with , the ramification index of at is a positive integer such that there is some open neighborhood of so that has only one preimage in , i.e., , and for all other points , . In other words, the map from to is to 1 except at . At all but finitely many points of , we have . Note that for any point we have . Sometimes the ramification index of at is called the valency of .

A set is said to be bounded from above if it has an upperbound.Consider the real numbers with their usual order. Then for any set , the supremum exists (in ) if and only if is bounded from above and nonempty.

A fiber of a map is the preimage of an element . That is,For instance, let and be the complex numbers . When , every fiber consists of two points , except for the fiber over 0, which has one point. Note that a fiber may be the empty set.In special cases, the fiber may be independent, in some sense, of the choice of . For instance, if is a covering map, then the fibers are all discrete and have the same cardinal number. The example is a covering map away from zero, i.e., from the punctured plane to itself has a fiber consisting of two points.When is a fiber bundle, then every fiber is isomorphic, in whatever category is being used. For instance, when is a real vector bundle of bundle rank , every fiber is isomorphic to .

A (symmetrical) boundary set of radius and center is the set of all points such thatLet be the origin. In , the boundary set is then the pair of points and . In , the boundary set is a circle. In , the boundary set is a sphere.

A point which is a member of the set closure of a given set and the set closure of its complement set. If is a subset of , then a point is a boundary point of if every neighborhood of contains at least one point in and at least one point not in .

A topology is given by a collection of subsets of a topological space . The smallest topology has two open sets, the empty set and . The largest topology contains all subsets as open sets, and is called the discrete topology. In particular, every point in is an open set in the discrete topology.

A Borel set is an element of a Borel sigma-algebra. Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable unions and intersections. Formally, the class of Borel sets in Euclidean is the smallest collection of sets that includes the open and closed sets such that if , , , ... are in , then so are , , and , where is a set difference (Croft et al. 1991).The set of rational numbers is a Borel set, as is the Cantorset.

A set is discrete in a larger topological space if every point has a neighborhood such that . The points of are then said to be isolated (Krantz 1999, p. 63). Typically, a discrete set is either finite or countably infinite. For example, the set of integers is discrete on the real line. Another example of an infinite discrete set is the set . On any reasonable space, a finite set is discrete. A set is discrete if it has the discrete topology, that is, if every subset is open.In the case of a subset , as in the examples above, one uses the relative topology on . Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this.

The tangent plane to a surface at a point is the tangent space at (after translating to the origin). The elements of the tangent space are called tangent vectors, and they are closed under addition and scalar multiplication. In particular, the tangent space is a vector space.Any submanifold of Euclidean space, and more generally any submanifold of an abstract manifold, has a tangent space at each point. The collection of tangent spaces to forms the tangent bundle . A vector field assigns to every point a tangent vector in the tangent space at .There are two ways of defining a submanifold, and each way gives rise to a different way of defining the tangent space. The first way uses a parameterization, and the second way uses a system of equations.Suppose that is a local parameterization of a submanifold in Euclidean space . Say,(1)where is the open unit ball in , and . At the point , the tangent space is the image of the Jacobian of , as a linear transformation..

A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in ). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.More concisely, any object that can be "charted" is a manifold.One of the goals of topology is to find ways of distinguishing manifolds. For instance, a circle is topologically the same as any closed loop, no matter how different these two manifolds may appear. Similarly,..

A (infinitely differentiable) manifold is said to be a submanifold of a manifold if is a subset of and the identity map of into is an embedding.

The closed graph theorem states that a linear operator between two Banach spaces and is continuous iff it has a closed graph, where the "graph" is considered closed if it is a closed subset of equipped with the product topology.The closed graph theorem also holds for Fréchetspaces.

Consider the set of compact -Riemannian manifolds with diameter, Volume, and where is the sectional curvature. Then there is a bound on the number of diffeomorphisms classes of this set in terms of the constants , , , and .

A lens space is the 3-manifold obtained by gluing the boundaries of two solid tori together such that the meridian of the first goes to a -curve on the second, where a -curve wraps around the longitude times and around the meridian times.

From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In an open set in there are no constraints, so the tangent space at a point is another copy of . The set could be a coordinate chart for an -dimensional manifold.The tangent space at , denoted , is the set of possible velocity vectors of paths through . Hence there is a canonical vector basis: if are the coordinates, then are a basis for the tangent space, where is the velocity vector of a particle with unit speed moving inward along the coordinate . The collection of all tangent vectors to every point on the manifold, called the tangent bundle, is the phase space of a single particle moving in the manifold .It seems as if the tangent space at is the same as the tangent space at all other points in the chart . However, while they do share the same dimension and are isomorphic, in a change of coordinates,..

The Stiefel manifold of orthonormal -frames in is the collection of vectors (, ..., ) where is in for all , and the -tuple (, ..., ) is orthonormal. This is a submanifold of , having dimension .Sometimes the "orthonormal" condition is dropped in favor of the mildly weaker condition that the -tuple (, ..., ) is linearly independent. Usually, this does not affect the applications since Stiefel manifolds are usually considered only during homotopy theoretic considerations. With respect to homotopy theory, the two definitions are more or less equivalent since Gram-Schmidt orthonormalization gives rise to a smooth deformation retraction of the second type of Stiefel manifold onto the first.

Any -dimensional Riemannian manifold can be locally embedded into an -dimensional manifold with Ricci curvature Tensor . A similar version of the theorem for a pseudo-Riemannian manifold states that any -dimensional pseudo-Riemannian manifold can be locally and isometrically embedded in an -dimensional pseudo-Euclidean space.

A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between the coordinate charts are smooth. A manifold with a smooth structure is called a smooth manifold (or differentiable manifold).A smooth structure is used to define differentiability for real-valued functions on a manifold. This extends to a notion of when a map between two differentiable manifolds is smooth, and naturally to the definition of a diffeomorphism. In addition, the smooth structure is used to define manifold tangent vectors, the collection of which is the tangent bundle.Two smooth structures are considered equivalent if there is a homeomorphism of the manifold which pulls back one atlas to an atlas compatible to the other one, i.e., a diffeomorphism. For instance, any two smooth structures on the circle are equivalent, as can be seen by integration.It is..

Another word for a (infinitely differentiable) manifold, also called a differentiable manifold. A smooth manifold is a topological manifold together with its "functional structure" (Bredon 1995) and so differs from a topological manifold because the notion of differentiability exists on it. Every smooth manifold is a topological manifold, but not necessarily vice versa. (The first nonsmooth topological manifold occurs in four dimensions.) Milnor (1956) showed that a seven-dimensional hypersphere can be made into a smooth manifold in 28 ways.

If is a differentiable homotopy sphere of dimension , then is homeomorphic to . In fact, is diffeomorphic to a manifold obtained by gluing together the boundaries of two closed -balls under a suitable diffeomorphism (Milnor).

Let be a map between two compact, connected, oriented -dimensional manifolds without boundary. Then induces a homomorphism from the homology groups to , both canonically isomorphic to the integers, and so can be thought of as a homomorphism of the integers. The integer to which the number 1 gets sent is called the degree of the map .There is an easy way to compute if the manifolds involved are smooth. Let , and approximate by a smooth map homotopic to such that is a "regular value" of (which exist and are everywhere dense by Sard's theorem). By the implicit function theorem, each point in has a neighborhood such that restricted to it is a diffeomorphism. If the diffeomorphism is orientation preserving, assign it the number , and if it is orientation reversing, assign it the number . Add up all the numbers for all the points in , and that is the , the Brouwer degree of . One reason why the degree of a map is important is because it is a homotopy invariant...

A relation between compact boundaryless manifolds (also called closed manifolds). Two closed manifolds are bordant iff their disjoint union is the boundary of a compact -manifold. Roughly, two manifolds are bordant if together they form the boundary of a manifold. The word bordism is now used in place of the original term cobordism.

The tangent space at a point in an abstract manifold can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called tangent vectors, and the collection of tangent spaces forms the tangent bundle.One description is to put an equivalence relation on smooth paths through the point . More precisely, consider all smooth maps where and . We say that two maps and are equivalent if they agree to first order. That is, in any coordinate chart around , . If they are similar in one chart then they are similar in any other chart, by the chain rule. The notion of agreeing to first order depends on coordinate charts, but this cannot be completely eliminated since that is how manifolds are defined.Another way is to first define a vector field as a derivation of the ring of smooth functions . Then a tangent vector at a point is an equivalence class of vector fields which agree at . That is, if for every smooth function . Of course,..

The only Wiedersehen surfaces are the standard round spheres. The conjecture was proven by combining the Berger-Kazdan comparison theorem with A. Weinstein's results for even and C. T. Yang's for odd. Green (1963) obtained the first proof of the Blaschke's conjecture in the two-dimensional case.

If is a closed oriented connected 3-manifold such that every simple closed curve in lies interior to a ball in , then is homeomorphic with the hypersphere, .

The study of manifolds having a complete Riemannian metric. Riemannian geometry is a general space based on the line elementwith for a function on the tangent bundle . In addition, is homogeneous of degree 1 in and of the form(Chern 1996). If this restriction is dropped, the resulting geometry is called Finsler geometry.

A type of abstract space which occurs in spline and rational function approximations. The Besov space is a complete quasinormed space which is a Banach space when , (Petrushev and Popov 1987).

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold (C-star) which is the one-point compactification of the complex numbers , together with two charts. (Here denotes complex infinity.) The notation is also used (Krantz 1999, p. 82; Lorentzen, and Waadeland 2008, p. 3).For all points in the complex plane, the chart is the identity map from the sphere (with infinity removed) to the complex plane. For the point at infinity, the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending infinity to 0 and all other points to .

On a compact oriented Finsler manifold without boundary, every cohomology class has a unique harmonic representation. The dimension of the space of all harmonic forms of degree is the th Betti number of the manifold.

Let be an open subset of the complex plane , and let denote the collection of all analytic functions whose complex modulus is square integrable with respect to area measure. Then , sometimes also denoted , is called the Bergman space for . Thus, the Bergman space consists of all the analytic functions in . The Bergman space can also be generalized to , where .

A map , between two compact Riemannian manifolds, is a harmonic map if it is a critical point for the energy functionalThe norm of the differential is given by the metric on and and is the measure on . Typically, the class of allowable maps lie in a fixed homotopy class of maps.The Euler-Lagrange differential equation for the energy functional is a non-linear elliptic partial differential equation. For example, when is the circle, then the Euler-Lagrange equation is the same as the geodesic equation. Hence, is a closed geodesic iff is harmonic. The map from the circle to the equator of the standard 2-sphere is a harmonic map, and so are the maps that take the circle and map it around the equator times, for any integer . Note that these all lie in the same homotopy class. A higher-dimensional example is a meromorphic function on a compact Riemann surface, which is a harmonic map to the Riemann sphere.A harmonic map may not always exist in a homotopy class,..

A vector field satisfying the vector identitywhere is the cross product and is the curl is said to be a Beltrami field.

Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category.Precisely stated, the theorem says that if a space is either a complete metric space or a locally compact T2-space, then the intersection of every countable collection of dense open subsets of is necessarily dense in .The above-mentioned interplay with first and second category sets can be summarized by a single corollary, namely that spaces that are either complete metric spaces or locally compact Hausdorff spaces are of second category in themselves. To see that this follows from the above-stated theorem, let be either a complete metric..

Let be a diffeomorphism on a compact Riemannian manifold . Then satisfies Axiom A if the nonwandering set of is hyperbolic and the periodic points of are dense in . Although it was conjectured that the first of these conditions implies the second, they were shown to be independent in or around 1977. Examples include the Anosov diffeomorphisms and Smale horseshoe map.In some cases, Axiom A can be replaced by the condition that the diffeomorphism is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and Pollicott 1990).

The Grassmannian is the set of -dimensional subspaces in an -dimensional vector space. For example, the set of lines is projective space. The real Grassmannian (as well as the complex Grassmannian) are examples of manifolds. For example, the subspace has a neighborhood . A subspace is in if and and . Then for any , the vectors and are uniquely determined by requiring and . The other six entries provide coordinates for .In general, the Grassmannian can be given coordinates in a similar way at a point . Let be the open set of -dimensional subspaces which project onto . First one picks an orthonormal basis for such that span . Using this basis, it is possible to take any vectors and make a matrix. Doing this for the basis of , another -dimensional subspace in , gives a -matrix, which is well-defined up to linear combinations of the rows. The final step is to row-reduce so that the first block is the identity matrix. Then the last block is uniquely determined by ...

An atlas is a collection of consistent coordinate charts on a manifold, where "consistent" most commonly means that the transition functions of the charts are smooth. As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like flat Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.The overlapping maps from one chart to another are called transition functions. They represent the transition from one chart's point of view to that of another. Let the open unit ball in be denoted . Then if and are two coordinate charts, the composition is a function defined on . That is, it is a function from an open subset of to , and given such a function from to , there are conditions for it to be smooth or have smooth derivatives (i.e., it is a C-k function). Furthermore, when is isomorphic to (in the even dimensional..

A special case of a flag manifold. A Grassmann manifold is a certain collection of vector subspaces of a vector space. In particular, is the Grassmann manifold of -dimensional subspaces of the vector space . It has a natural manifold structure as an orbit-space of the Stiefel manifold of orthonormal -frames in . One of the main things about Grassmann manifolds is that they are classifying spaces for vector bundles.

Given a compact manifold and a transversely orientable codimension-one foliation on which is tangent to , the pair is called a generalized Reeb component if the holonomy groups of all leaves in the interior are trivial and if all leaves of are proper. Generalized Reeb components are obvious generalizations of Reeb components.The introduction of the generalized version of the Reeb component facilitates the proof of many significant results in the theory of 3-manifolds and of foliations. It is well-known that generalized Reeb components are transversely orientable and that a manifold admitting a generalized Reeb component also admits a nice vector field (Imanishi and Yagi 1976). Moreover, given a generalized Reeb component , is a fibration over .Like many notions in geometric topology, the generalized Reeb component can be presented in various contexts. One source describes a generalized Reeb component on a closed 3-manifold with foliation..

Two closed simply connected 4-manifolds are homeomorphic iff they have the same bilinear form and the same Kirby-Siebenmann invariant . Any can be realized by such a manifold. If is odd for some , then either value of can be realized also. However, if is always even, then is determined by , being congruent to 1/8 of the signature of . Here, is a symmetric bilinear form with determinant (Milnor).In particular, if is a homotopy sphere, then and , so is homeomorphic to .

A manifold with a Riemannian metric that has zero curvature is a flat manifold. The basic example is Euclidean space with the usual metric . In fact, any point on a flat manifold has a neighborhood isometric to a neighborhood in Euclidean space. A flat manifold is locally Euclidean in terms of distances and angles, as well as merely topologically locally Euclidean, as all manifolds are.The simplest nontrivial examples occur as surfaces in four dimensional space. For instance, the flat torus is a flat manifold. It is the image of . A theorem due to Bieberbach says that all compact flat manifolds are tori. More generally, the universal cover of a complete flat manifold is Euclidean space.

An ambient isotopy from an embedding of a manifold in to another is a homotopy of self diffeomorphisms (or isomorphisms, or piecewise-linear transformations, etc.) of , starting at the identity map, such that the "last" diffeomorphism compounded with the first embedding of is the second embedding of . In other words, an ambient isotopy is like an isotopy except that instead of distorting the embedding, the whole ambient space is being stretched and distorted and the embedding is just "coming along for the ride." For smooth manifolds, a map is isotopic iff it is ambiently isotopic.For knots, the equivalence of manifolds under continuous deformation is independent of the embedding space. Knots of opposite chirality have ambient isotopy, but not regular isotopy.

For any sequence of integers , there is a flag manifold of type (, ..., ) which is the collection of ordered sets of vector subspaces of (, ..., ) with and a subspace of . There are also complex flag manifolds with complex subspaces of instead of real subspaces of a real -space.These flag manifolds admit the structure of manifoldsin a natural way and are used in the theory of Lie groups.

A hypersphere is parallelizable if there are vector fields that are linearly independent at each point. There exist only three parallelizable spheres: , , and (Adams 1958, 1960, Le Lionnais 1983).More generally, an -dimensional manifold is parallelizable if its tangent bundle is a trivial bundle (i.e., if is globally of the form ).

The real projective plane with elliptic metric where the distance between two points and is defined as the radian angle between the projection of the points on the surface of a sphere (which is tangent to the plane at a point ) from the antipode of the tangent point.

The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The strip bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).The Möbius strip has Euler characteristic (Dodson and Parker 1997, p. 125).According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. M. C. Escher was fond of portraying Möbius strips, and..

The generalization of the Schönflies theorem to dimensions. A smoothly embedded -hypersphere in an -hypersphere separates the -hypersphere into two components, each homeomorphic to -balls. It can be proved using Morse theory.

Thurston's conjecture proposed a complete characterization of geometric structureson three-dimensional manifolds.Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."Thurston's conjecture is that, after you split a three-manifold into its connected sum and..

Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient , where is a finite group of rotations that acts freely on .Since the trivial group is in particular a finite group, the elliptization conjecture implies the Poincaré conjecture.

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