Biquandle virtual brackets

Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. It was shown by Makowsky in 2001 that computing the Jones polynomial is fixed-parameter tractable in the treewidth of the link diagram, but the parameterised complexity of the more powerful HOMFLY-PT polynomial remained an open problem. Here we show that computing HOMFLY-PT is fixed-parameter tractable in the treewidth, and we give the first sub-exponential time algorithm to compute it for arbitrary links.

Topological quantum field theories provide some of the most interesting examples for the usefulness of path integrals. One of the best known of these examples was discovered in [32] where one particular topological quantum field theory, Chern-Simons theory, was studied and the so-called “Wilson loop observables” (WLOs) were computed explicitly. These WLOs are heuristic path integrals and the interesting thing about the expressions obtained in [32] is that they involve highly non-trivial link invariants like the Jones polynomial, the HOMFLY polynomial, and the Kauffman polynomial, cf. [26, 161. A more thorough study of the WLOs by other methods [18, 8, 5, 171 later led to a breakthrough in knot theory, the discovery of the universal Vassiliev invariant [27]. Unfortunately, it has not yet been possible to establish the aforementioned connection between path integrals and knot polynomials at a rigorous level. In the special case, however, where the base manifold M of the Chern-Simons model considered is of product form the situation looks much more promising and as we will show in the present paper it is reasonable to expect that, at least for some of these special manifolds M, it will eventually be possible

We prove that the number of linearly independent Vassiliev invariants for an r-component link of order n, which derived from the HOMFLY polynomial, is greater than or equal to min{n, [(n+ r − 1)/2]}. Introduction. Let Vn denote the vector space consisting of all Vassiliev knot invariants of order less than or equal to n. There is a filtration V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn ⊂ · · · in the entire space of Vassiliev knot invariants. Each Vn is finite-dimensional. Vassiliev [V] studied for the special cases when n is small: V0 = V1, which consists of a constant map (Propositions 3 and 5), and V2/V1 is a one-dimensional vector space, whose basis is the second coefficient of the Conway polynomial. The dimensions for small n are found by using the computer by Bar-Natan and Stanford (cf. [BN; B1, p. 282 ]): For n = 1, 2, 3, 4, 5, 6, 7, dimVn/Vn−1 = 0, 1, 1, 3, 4, 9, 14, respectively. On the other hand, Bar-Natan (cf. [BN]) showed that the nth coefficient of the Conway polynomial is of order less than or equal to n. Birman and Lin [BL] and Gusarov [G] proved that the Jones, HOMFLY, and Kauffman polynomials of a knot can be interpreted as an infinite sequence of Vassiliev knot invariants, and as a corollary they proved that dimVn/Vn−1 ≥ 1 for every n ≥ 2 using the HOMFLY polynomial [BL, Corollary 4.2 (i)]. 1991 Mathematics Subject Classification: Primary 57M25.

We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M, L), where M is a closed oriented 3-manifold, and is an oriented link, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in having the same HOMFLY polynomial and the same Kauffman polynomial but distinct Turaev-Viro invariants, and similar examples for the Alexander polynomial. We also investigate the relations between the Turaev-Viro invariants of (M, L) and those of , showing that they are sometimes, but not always, determined by each other, and discuss some relations with the Witten-Reshetikhin-Turaev invariants and the Jones polynomial.

Author(s): Rutherford, Dan | Abstract: We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar methods a result involving the upper bound given by the HOMFLY polynomial and 2-graded rulings is proved.

We recall first some known facts on Jones and Kauffman polynomials for links, and on state models for link invariants. We give next an exposition of a recent spin model due to F. Jaeger and which involves Higman-Sims graph. The associated invariant assigns to an oriented link evaluation for a = -τ 5 and z = 1 of its Kauffman polynomial in Dubrovnik form, where τ denotes golden ratio. 1. Introduction. A knot is a simple closed curve in R 3 and a link is a finite union of disjoint knots. We denote by L a link L together with an orientation on each of its components. Two oriented links L, L' are isotopic, and we write L' « L, if there exists a family (Φt)o<t< of homeomorphism s of R3 such that map [0, 1] —• R3 sending t to φt(x) is continuous for each x e R3 and such that φo = id, φ\(L) = L 1, where last equation indicates that orientations correspond via φ. Links considered here are always assumed to be tame, namely isotopic to links made of smoothly embedded curves. A Ω-valued invariant for oriented links is a map L »-• I(L) which associates to each oriented link L in R3 an element I(L) of some ring Ω, for example C or a ring of Laurent polynomials, in such a way that I(Lf) = I(L) whenever Lf « L. Classically, one of most studied example of link invariant is Alexander-Conway polynomial Δ(L) e Z[t±ι] defined by J. W. Alexander in 1928 [Ale], with a normalization made precise by J. H. Conway in 1969 [Con]; notation (L rather than L) indicates that, at least for knots, Δ(L) does not depend on choice of an orientation on knot. The polynomial invariant L —• Δ(L) is well understood in terms of standard algebraic topology (homology of the infinite cyclic covering of complement of L in R 3) see e.g. [Rha], [Rol] or [BuZ]. The subject entered a new era in 1984 [Jol] with discovery of Jones polynomial V(L) e Z[t±ι/2]. This was starting point of several other invariants, including Kauffman polynomial

Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width

We consider vector spaces \mathcal H_{n,\ell} and \mathcal F_{n,\ell} spanned by the degree- n coefficients in power series forms of the Homfly and Kauffman polynomials of links with \ell components. Generalizing previously known formulas, we determine the dimensions of the spaces \mathcal H_{n,\ell}, \mathcal F_{n,\ell} and \mathcal H_{n,\ell}+\mathcal F_{n,\ell} for all values of n and \ell . Furthermore, we show that for knots the algebra generated by \bigoplus_n \mathcal H_{n,1}+\mathcal F_{n,1} is a polynomial algebra with \dim(\mathcal H_{n,1}+\mathcal F_{n,1})-1=n+[n/2]-4 generators in degree n\geq 4 and one generator in degrees 2 and 3.

We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra G2. It is therefore related to G2 in the same way that the HOMFLY polynomial is related to An and the Kauffman polynomial is related to Bn, Cn, and Dn. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions.

Bracket models for weight systems and the universal Vassiliev invariants

We discuss new methods of changing knot and link diagrams in such a way that their Jones polynomial is unaltered. It is similar to Conway's mutation, but applies to more complicated "tangles," and is performed simultaneously at multiple sites in the diagram — thus "global mutation." The method of achieving this can be regarded as communication via "messenger tangles."

By the use of the Karoubi Tower diagram we generalize the classical invariants of quadratic forms. Similar to Quillen's higher K-theory generalization of the classical K-theory groups, these invariants are an extension of the classical invariants by the use of homotopy theory. The iterated forgetful maps in the Karoubi Tower are KR valued and yield a generalization of the standard (rank, discriminant and total Hasse-Witt) invariants of quadratic forms in two directions. First, we get invariants of all degrees. Second, these invariants are defined for every Hermitian ring. They yield and generalize the Clifford invariant in the case of a field of characteristic different from 2, or in the case of an arithmetic Dedekind domain containing 1 . 1 and K2=2K2 valued, respectively. We write Ki for the ith K- theory group KiR. These invariants assemble into the Clifford invariant, which is a homomorphism from the Witt ring W.R/into the Brauer-Wall group BW.R/ .I n some special cases, for example in the case of a finite field Fpi , in the case of p-adics b Qp, or in the case of algebraic number fields, the Clifford invariant (together with rank and signatures) classify the quadratic forms (see (16)). Recall that when 1 2 R the category of quadratic forms and the category of Hermitian forms over .R; id; 1/ are equivalent. The classical invariants of quadratic forms over a field were generalized in many ways. In particular, one desires to have a general theory of quadratic forms over any commutative ring (see (2, 8)). For another example of generalized classical invariants see Giffen's K2 valued Hasse-Witt invariants for .; /-reflexive forms (4). In (11), Milnor defines 'Stiefel-Whitney invariants' of quadratic forms over a field. These

Character expansion expresses extended HOMFLY polynomials through traces of products of finite-dimensional [Formula: see text]- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding [Formula: see text]-matrices. Even a weaker (and, perhaps, more reliable) version of this conjecture is sufficient to explicitly calculate HOMFLY polynomials for all the 3-strand braids in arbitrary (anti)symmetric representations. We list the examples of so obtained polynomials for R = [3] and R = [4], and they are in accordance with the known answers for torus and figure-eight knots, as well as for the colored special and Jones polynomials. This provides an indirect evidence in support of our conjecture.

In this paper, we show some relations for the HOMFLYPT and Jones polynomials among certain families of amphicheiral 2-bridge knots and links.

We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials when evaluated on classical links. Further, the extension of the Kauffman polynomial is more powerful of the Homflypt polynomial, as well as of certain new invariants introduced recently. Also we propose a new algebra which plays in the case of tied links the same role as the BMW algebra for the Kauffman polynomial in the classical case. Moreover, we prove that the Markov trace on this new algebra can be recovered from the extension of the Kauffman polynomial defined here.