Differential expansion and rectangular HOMFLY for the figure eight knot

TL;DR

This paper develops a conjecture for the differential expansion of the rectangularly colored HOMFLY polynomial of the figure eight knot, based on known results for the trefoil and the properties of differential expansion.

Contribution

It introduces a plausible conjecture for the rectangular HOMFLY of the figure eight knot, extending previous results for symmetric and antisymmetric representations.

Findings

Conjecture for rectangular HOMFLY of the figure eight knot.

Relation between DE for trefoil and figure eight knots.

Extension of known results to all rectangular representations.

Abstract

Differential expansion (DE) for a Wilson loop average in representation $R$ is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of $3d$ Chern-Simons theory. Especially simple is the relation between the DE for the trefoil $3_1$ and for the figure eight knot $4_1$. Since arbitrary colored HOMFLY for the trefoil are known from the Rosso-Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfil this program for all rectangular representation $R=[r^s]$, i.e. make a plausible conjecture for the rectangularly colored HOMFLY of the figure eight knot, which generalizes the old result for totally symmetric and antisymmetric representations.