Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid

TL;DR

This paper introduces a systematic character expansion method for HOMFLY polynomials of knots, simplifying their computation and revealing a hierarchical structure in the expansion coefficients, applicable to braids with up to five strands.

Contribution

It presents a new explicit character expansion approach for HOMFLY polynomials, enabling direct calculation in terms of A=q^N and uncovering a hierarchical structure in the coefficients.

Findings

Expansion coefficients depend on braid realization but not on knot invariants.

Method simplifies HOMFLY polynomial calculations compared to skein relations.

Hierarchical structure allows expressing coefficients through elementary constituents.

Abstract

Character expansion is introduced and explicitly constructed for the (non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients are not the knot invariants and can depend on the choice of the braid realization. However, the method provides the simplest systematic way to construct HOMFLY polynomials directly in terms of the variable A=q^N: a much better way than the standard approach making use of the skein relations. Moreover, representation theory of the simplest quantum group SU_q(2) is sufficient to get the answers for all braids with m<5 strands. Most important we reveal a hidden hierarchical structure of expansion coefficients, what allows one to express all of them through extremely simple elementary constituents. Generalizations to arbitrary knots and arbitrary representations is straightforward.