On rectangular HOMFLY for twist knots

TL;DR

This paper advances the understanding of rectangular HOMFLY knot polynomials by reformulating their computation for twist knots using skew Schur polynomials, highlighting new challenges in generalization.

Contribution

It introduces a reformulation of rectangular HOMFLY polynomials for twist knots in terms of skew Schur polynomials, revealing complexities in extending to arbitrary rectangular representations.

Findings

Reformulation in terms of skew Schur polynomials

Identification of shifts from the standard topological locus

Challenges in generalizing to arbitrary rectangular representations

Abstract

As a new step in the study of rectangularly-colored knot polynomials, we reformulate the prescription of arXiv:1606.06015 for twist knots in the double-column representations $R=[rr]$ in terms of skew Schur polynomials. These, however, are mysteriously shifted from the standard topological locus, what makes further generalization to arbitrary $R=[r^s]$ not quite straightforward.