On colored HOMFLY polynomials for twist knots

TL;DR

This paper advances the computation of colored HOMFLY polynomials for twist knots by applying the evolution method to non-rectangular representations, bridging a gap in knot invariant calculations.

Contribution

It introduces a deformation of the differential expansion and applies the evolution method to non-rectangular representations for twist knots, enabling broader calculations.

Findings

Initial conditions for H_{[21]} polynomials are established.

The deformation of the differential expansion is identified.

New insights for generic representations in twist knots are suggested.

Abstract

Recent results of J.Gu and H.Jockers provide the lacking initial conditions for the evolution method in the case of the first non-trivially colored HOMFLY polynomials H_{[21]} for the family of twist knots. We describe this application of the evolution method, which finally allows one to penetrate through the wall between (anti)symmetric and non-rectangular representations for a whole family. We reveal the necessary deformation of the differential expansion, what, together with the recently suggested matrix model approach gives new opportunities to guess what it could be for a generic representation, at least for the family of twist knots.