Kauffman type invariants for tied links

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

In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent by mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.

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

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 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.

We prove that the Framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. We use this result to obtain a closed combinatorial formula for the invariants for classical links obtained from a Markov trace on the Framisation of the Temperley–Lieb algebra. For a given link L, this formula involves the Jones polynomials of all sublinks of L, as well as linking numbers.

Introduction. In [Jon83], Jones constructs, from any III subfactor pair M,N with finite index [M, N] = q + q-I + 2, a canonical hyperfinite subfactor pair of the same index. The pair is constructed from the completion of the TemperleyLieb algebra by a Markov trace. By showing that such a completion yields a II, subfactor pair exactly when q is a positive real or of the form e27ri/k, he obtained his index of subfactors result. The fact that the set of admissible values of q divides into a discrete piece and a continuous piece is puzzling and is indicative of qualitative differences between the subfactors themselves. For one thing, at roots of unity the subfactors are of finite depth. That is, after a certain point in the Bratteli diagram the centers of the finite-dimensional approximating algebras are periodically isomorphic. On the contrary, in the positive real case each finite-dimensional algebra in the Bratteli diagram has a strictly larger center than the previous one. Furthermore, the relative commutant of the smaller algebra in the larger algebra consists only of multiples of the identity in the root of unity case. For positive real q, not only is there a relative commutant, but in fact the subfactor is locally trivial. That is, the subfactor pair, restricted to a minimal projection in the relative commutant, is the trivial subfactor pair. In addition to its direct relevance to the study of subfactors, Jones' work had indirect but important ramifications. The Temperley-Lieb algebra in the construction forms a representation of the braid group, and the Markov trace as a functional on the braid group is, with proper normalization, a polynomial link invariant. This discovery triggered an explosion of mathematics which eventually came to see the algebra, trace, and invariant as a special case of a general construction which associates, via quantum groups, an example of all three to each irreducible representation of each. simple Lie algebra. The original Jones polynomial comes from the fundamental representation of s12. The quantum groups construction can be done entirely without reference to subfactors, or even analysis. Nevertheless, there are echos of Jones' work, and in particular both the roots of unity and the positive real q continue to play a distinct and special role. For example, when q is a root of unity, the quantum group

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring [Formula: see text] using virtual crossings as smoothings, these invariants take the form of multisets of elements of [Formula: see text] and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic elementary diagrams each of which is associated to a matrix that yields solutions of the quantum Yang-Baxter equation. We recover the bracket polynomial, and define the rotational bracket polynomial, the binary bracket polynomial, the Alexander polynomial, the generalized Alexander polynomial and an infinity of specializations of the Homflypt polynomial for Morse knotoids via quantum state sum models.

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.

Framization of a Temperley-Lieb algebra of type B

Bracket models for weight systems and the universal Vassiliev invariants

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.