Related papers: An expression for the Homflypt polynomial and some…
In this paper, I give a method to calculate the HOMFLY polynomials of knots by using a representation of the braid group B4 into a group of 3 ? 3 matrices. Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same…
The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a…
Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent…
We will prove that, for a $2$ or $3$ component $L$-space link, $HFL^-$ is completely determined by the multi-variable Alexander polynomial of all the sub-links of $L$, as well as the pairwise linking numbers of all the components of $L$. We…
We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyze relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the…
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…
Khovanov has given a construction of the Khovanov-Rozansky link invariants (categorifying the HOMFLYPT invariant) using Hochschild cohomology of 2-braid groups. We give a direct proof that his construction does give link invariants. We show…
We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann…
The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the…
Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial…
We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of $(n,m)$ torus knots, and present a number of equivalent expressions, all related…
We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation…
This short note is about three-stranded pretzel knots that have an even number of crossings in one of the strands. We calculate the braid index of such knots and determine which of them are quasipositive. The main tools are the…
We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these…
We give conditions on a knot on which the Morton-Franks-Williams inequality is not sharp. As applications, we show infinitely many examples of knots where the inequality is not sharp and also prove (by giving examples) that the deficit of…
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of…
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed…
The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to…
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 give the first known topological model for the HOMFLY-PT polynomial constructed directly from link diagrams. More precisely, we prove that this invariant is given by graded intersections between explicit Lagrangian submanifolds in a…