Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux

TL;DR

This paper presents a new method to compute colored HOMFLY polynomials using sums over paths or standard Young tableaux, simplifying calculations by exploiting the structure of R-matrices in braid representations.

Contribution

It introduces a path-based and Young tableaux approach to evaluate colored HOMFLY polynomials, making the process more efficient and explicit for arbitrary braids.

Findings

Path sum formula simplifies R-matrix calculations.

R-matrices reduce to 1x1 and 2x2 blocks with explicit expressions.

Method is effective for evaluating colored HOMFLY polynomials for arbitrary braids.

Abstract

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 braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity M_{RQ} of Q in R^{\otimes m}. If R is the fundamental representation R=[1], then M_{[1] Q} is equal to the number of paths in representation graph, which lead from the fundamental vertex [1] to the vertex Q. In the basis of paths the entries of the m-1 relevant R-matrices are associated with the pairs of paths and are non-vanishing only when the two paths either coincide or differ by at most one vertex; as a corollary R-matrices consist of just 1x1 and 2x2 blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation R, then the braid has m|R| strands, Q have a bigger size m|R|, but only paths passing through the vertex R are included into the sums over paths which define the products and traces of the m|R|-1 relevant R-matrices. In the case of SU(N) this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids.