Related papers: On the HOMFLY and Tutte polynomials
Let $t_{i,j}$ be the coefficient of $x^iy^j$ in the Tutte polynomial $T(G;x,y)$ of a connected bridgeless and loopless graph $G$ with order $n$ and size $m$. It is trivial that $t_{0,m-n+1}=1$ and $t_{n-1,0}=1$. In this paper, we obtain…
The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid $N$, we associate…
The Tutte polynomial of a graph, or equivalently the $q$-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this…
The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of…
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the…
We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual $q$ and $A = q^N$. These parameters preserve topological invariance and do not…
Can a non self-complementary graph have the same chromatic polynomial as its complement? The answer to this question of Akiyama and Harrary is positive and was given by J. Xu and Z. Liu. They conjectured that every such graph has the same…
There are several different extensions of the Tutte polynomial to graphs embedded in surfaces. To help frame the different options, here we consider the problem of extending the Tutte polynomial to cellularly embedded graphs starting from…
Khovanov--Rozansky's HOMFLY homology is determined for all prime knots with up to 11 crossings by direct computations.
Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points (a,b) where (a-1)(b-1) equals 2 or 4. These evaluations are expressed in terms of…
We introduce a certain class of link diagrams, which includes all closed braid diagrams. We show a generalized version of K\'alm\'an's full-twist formula for the HOMFLY polynomial in the class.
We combinatorially prove a new recurrence between the Tutte polynomials of graphs obtained by contraction of the complete graphs $K_{n}$%. This generalizes, to two variables, a relation previously obtained by the author between the…
Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by…
T. K\'{a}lm\'{a}n (A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1/x,1)$ and…
Recently, it has been shown that the Jones polynomial, in [LS19], and the Alexander polynomial, in [NT18], of rational knots can be obtained by specializing $F$-polynomials of cluster variables. At the core of both results are continued…
This article computes the Tutte polynomial of the hyperplane arrangements associated to the complex reflection groups. The calculations are based on both formulas of De Concini and Procesi for Tutte polynomial and the normaliser of…
We generalize the recently discovered planar decomposition (Kauffman bracket) for the HOMFLY polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after…
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY…
We compare the invariant for classical knots and links defined using the Juyumaya trace on the Yokonuma-Hecke algebras with the HOMFLYPT polynomial. We show that the two invariants, as maps on the set ${\mathcal L}$ of oriented link types…
The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point…