Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
A.Anokhina, A.Mironov, A.Morozov, An.Morozov

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…
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