Factorization of differential expansion for non-rectangular representations

TL;DR

This paper extends the factorization of differential expansion coefficients for HOMFLY-PT polynomials from rectangular to certain non-rectangular representations, proposing a conjecture for all representations of the form [r,1] and analyzing the structure of Racah matrices.

Contribution

It generalizes the factorization property of differential expansion to non-rectangular representations and provides a conjecture for all [r,1] cases, advancing understanding of knot polynomial structures.

Findings

Factorization extends to [2,1] and [3,1] representations.

Conjecture for double-braid polynomials for all [r,1] representations.

Partial extraction of Racah matrix elements and unitarity-based reconstruction.

Abstract

Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations $R$, is extended to the first non-rectangular representations $R=[2,1]$ and $R=[3,1]$. This increases chances that such factorization will take place for generic $R$, thus fixing the shape of the DE. We illustrate the power of the method by conjecturing the DE-induced expression for double-braid polynomials for all $R=[r,1]$. In variance with rectangular case, the knowledge for double braids is not fully sufficient to deduce the exclusive Racah matrix $\bar S$ -- the entries in the sectors with non-trivial multiplicities sum up and remain unseparated. Still a considerable piece of the matrix is extracted directly and its other elements can be found by solving the unitarity constraints.