相关论文: A bicomplex of Khovanov homology for colored Jones…

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of…

We discuss two realizations of the colored Jones polynomials of a knot, one from an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another one from…

The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the…

We extend the definition of the colored Jones polynomials to framed links and trivalent graphs in S^3 # k S^2 X S^1 using a state-sum formulation based on Turaev's shadows. Then, we prove that the natural extension of the Volume Conjecture…

This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we…

We consider s-chromatic polynomials of simplicial complexes, higher dimensional analogues of chromatic polynomials for graphs.

Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the…

Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic…

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled…

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are $q$-holonomic, that is, they satisfy linear $q$-difference equations with coefficients Laurent…

The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to…

The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the…

We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of…

The "color" in the colored Jones polynomial is an integer parameter. In this paper, a periodic pattern of the values of the colored Jones polynomial at the second and the third roots of unity is found. If we substitute -1 to the colored…

We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how…

We construct an endomorphism of the Khovanov invariant to prove H-thinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is…

In this paper we compute Hochschild homology of certain Soergel bimodules. Moreover, we describe explicitly the graded bimodule maps between Soergel bimodules. This computations are motivated by the categorifications of the colored…

We construct a well-defined relative second grading on symplectic Khovanov cohomology from holomorphic disc counting. We show that it recovers the Jones grading of Khovanov homology up to an overall grading shift over any characteristic…

The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks…

In the cabling procedure for HOMFLY polynomials colored HOMFLY polynomials of a knot are obtained from ordinary HOMFLY of the cabled knot with extra twists added. Thus colored polynomials can be seen as relation between HOMFLYs of cabled…