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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).

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).

An orientable surface with one boundary component such that the boundary component of the surface is a given knot . In 1934, Seifert proved that such a surface can be constructed for any knot. The process of generating this surface is known as Seifert's algorithm. Applying Seifert's algorithm to an alternating projection of an alternating knot yields a Seifert surface of minimal knot genus.There are knots for which the minimal genus Seifert surface cannot be obtained by applying Seifert's algorithm to any projection of that knot, as proved by Morton in 1986 (Adams 1994, p. 105).

The figure eight knot, also known as the Flemish knot and savoy knot, is the unique prime knot of four crossings 04-001. It has braid word .The figure eight knot is implemented in the WolframLanguage as KnotData["FigureEight"].It is a 2-embeddable knot, and is amphichiral as well as invertible. It has Arf invariant 1. It is not a slice knot (Rolfsen 1976, p. 224).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the figure eight knot are(1)(2)(3)(4)(5)(6)There are no other knots on 10 or fewer crossings sharing the same Alexander polynomial, BLM/Ho polynomial, bracket polynomial, HOMFLY polynomial, Jones polynomial, or Kauffman polynomial F.The figure eight knot has knot group(7)(Rolfsen 1976, p. 58).Helaman Ferguson's sculpture "Figure-Eight Complement II" illustrates the knot complement of the figure eight..

Consider strings, each oriented vertically from a lower to an upper "bar." If this is the least number of strings needed to make a closed braid representation of a link, is called the braid index. A general -braid is 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 .The operations and on strings define a group known as the braid group or Artin braid group, denoted .Topological equivalence for different representations of a braid word and is guaranteed by the conditions(1)as first proved by E. Artin.Any -braid can be expressed as a braid word, e.g., is a braid word in the braid group . When the opposite ends of the braids are connected by nonintersecting lines, knots (or links) may formed that can be labeled by their corresponding..

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