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

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.

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity.

Zeros of Jones polynomials for families of knots and links

The behavior of Tutte polynomials of graphs under five graph operations and its applications

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

A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan

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.

Using a variation of the interpretability concept we show that all graph properties definable in monadic second order logic (MS properties) with quantification over vertex and edge sets can be decided in linear time for classes of graphs of fixed bounded tree-width, giving an alternative proof of a recent result by Courcelle. We allow graphs with directed and/or undirected edges, labeled on edges and/or vertices with labels taken from a finite set. We extend MS properties to Extended Monadic Second-order (EMS) problems involving counting or summing evaluations given with the graph over sets definable in monadic second order logic. Our tecnique allowes us to solve also some EMS problems in linear time or in polynomial or pseudopolynomial time for classes of graphs of fixed bounded tree-width. Most problems for wich linear time algorithms for graphs of bounded tree width where previously known to exist, and many others, are EMS problems.

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.

A permanent formula for the Jones polynomial

A Graph G=(V,E) is called k-slim if for every subgraph S=(VS,ES) of G with s=|VS|≥k there exists K⊂VS, |K|=k, such that the vertices of VS\K can be partitioned into two subsets, A and B, such that |A|≤⅔s and |B|≤⅔s and no edge of ES connects a vertex from A and a vertex from B. k-slim graphs contain, in particular, the graphs with tree-width k. In this paper we give an algorithm solving the H-decomposition problem for a large family of graphs H which contains, among others, the stars, the complete graphs, and the complete r-partite graphs where r≥3. The algorithm runs in polynomial time in case the input graph is k-slim, where k is fixed. In particular, our algorithm runs in polynomial time when the input graph has bounded tree width k. Our results supply the first polynomial time algorithm for H-decomposition of connected graphs H having at least 3 edges, in graphs with bounded tree-width.

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.

Bodlaender's Theorem states that for every k there is a linear-time algorithm that decides whether an input graph has tree width k and, if so, computes a width-k tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula φ and for every k there is a linear-time algorithm that decides whether a given logical structure A of tree width at most k satisfies φ. We prove that both theorems still hold when linear time is replaced by space. The transfer of the powerful theoretical framework of monadic second-order logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the log space world.

Randomized online graph coloring