Narrow Results

We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing mutants. Then we study the distinction of mutants, with particular regard to the symmetric mutants. We discuss the determination of braid index as another application of our computational methods.

We give a basis for the space of the Vassiliev knot invariants of order 6, which implies a sixth order Vassiliev knot invariant is determined by its HOMFLY and Kauffman polynomials. Furthermore, we show that there is a seventh order invariant which cannot be obtained from these polynomials of a knot.

We show that the integer-valued knot invariants appearing as the nontrivial coefficients of the HOMFLY polynomial, the Kauffman polynomial and the Q-polynomial are not of finite type.

Following the recent work by Chan [1] and Morton and Hadji [2] on the Homflypt polynomials of some generalized Hopf links, we investigate the polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S¹ × D², we establish a similar skein map on the Kauffman skein module of S¹ × D² which has distinct eigenvalues. Furthermore we are able to calculate the Kauffman polynomials of some specific generalized Hopf links.

In the present paper, we construct a virtual knot invariant with values in the free infinite-dimensional module over Z[a, a^-1]. The restriction of this invariant to the set of classical knots coincides with the Jones–Kauffman polynomial. It distinguishes virtual knots stronger than the generalised Jones-Kauffman polynomial proposed in [Kau].

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S³ along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. -moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

We exhibit an infinite family of knots that are detected chiral by the Kauffman polynomial but not by the HOMFLY polynomial.

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 2-variable polynomials, answering a question raised by Dunfield et al. in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v = s², we give examples whose Homfly polynomials differ when v = s³. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots.

This paper aims to determine the images of the braid group under representations afforded by the Yang–Baxter equation when the solution is a non-trivial 4 × 4 matrix. Making the assumption that all the eigenvalues of the Yang–Baxter solution are roots of unity, leads to the conclusion that all the images are finite.

Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant.

This paper consists of three parts.

First, we generalize the Jaeger Formula to express the Kauffman–Vogel graph polynomial as a state sum of the Murakami–Ohtsuki–Yamada graph polynomial.

Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the 𝔰𝔩(N) Murakami–Ohtsuki–Yamada polynomial or the 𝔰𝔩(N) homology of this MOY graph. In fact, reversing the orientation and the color of a component of a colored link only changes the 𝔰𝔩(N) homology by an overall grading shift.

Finally, as an application of the first two parts, we prove that the 𝔰𝔬(6) Kauffman polynomial is equal to the 2-colored 𝔰𝔩(4) Reshetikhin–Turaev link polynomial, which implies that the 2-colored 𝔰𝔩(4) link homology categorifies the 𝔰𝔬(6) Kauffman polynomial.

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

We study the relation between the maximum z-degree of Kauffman polynomial and the determinant for a quasi-alternating link. This gives a stronger criterion for deciding whether a non-alternating link is quasi-alternating or not than the known criterion in terms of Q-polynomials. Also, we determine all quasi-alternating links with determinant 5.

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of $2$-cable links and Whitehead doubles of all prime knots with up to eight crossings.

It is known that every knot bounds a singular disk with only clasp singularities, which is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. It is known that the Conway polynomials of knots with clasp number at most two are characterized. In this paper, we focus on the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials, which is called the $\mathrm{\Gamma }$-polynomial. As a result, we characterize the $\mathrm{\Gamma }$-polynomials of knots with clasp number at most two. Moreover, it is known that there exist two homeomorphic classes of clasp disks with two clasp singularities, which are called types $0$ and $1$. We show that there exist infinitely many knots which bound clasp disks of not doubled knots but both types $0$ and $1$, not type $0$ but type $1$, not type $1$ but type $0$, respectively.

In this paper, we calculate the Kauffman polynomials $F(K(p,q);a,z)$ of Kanenobu knots $K(p,q)$ with $p,q$ half twists and determine their spans on the variable $a$ completely. As an application, we determine the arc index of infinitely many Kanenobu knots. In particular, we give sharper lower bounds of the arc index of $K(2n,-2n)$ by using canonical cabling algorithm and the 2-cable $\mathrm{\Gamma }$-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial of a knot is the $\mathrm{\Gamma }$-polynomial of the $(p,q)$-cable knot of the knot, where the $\mathrm{\Gamma }$-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial $(2,1)$-cable $\mathrm{\Gamma }$-polynomial, that is, the $(2,1)$-cable $\mathrm{\Gamma }$-polynomial of the trivial knot. Moreover, we see that the knots have the trivial $\mathrm{\Gamma }$-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.

For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial of a knot is the $\mathrm{\Gamma }$-polynomial of the $(p,q)$-cable knot of the knot, where the $\mathrm{\Gamma }$-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the $\mathrm{\Gamma }$-polynomial is computable in polynomial time, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial is also computable in polynomial time. In this paper, we show that the $(2,1)$-cable $\mathrm{\Gamma }$-polynomial completely classifies the unoriented knots up to ten crossings including the chirality information.

We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, $H[R]$, $K[Q]$ and $D[T]$, based on the invariants of knots, $R$, $Q$ and $T$, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants ($R$, $Q$, $T$) on sublinks of a given link $L$, obtained by partitioning $L$ into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.