Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation

TL;DR

This paper derives a general formula for extended HOMFLY polynomials of 3-strand braids in the first symmetric representation, using character decompositions and symmetry considerations, advancing knot invariant calculations.

Contribution

It provides a new explicit formula for extended HOMFLY polynomials for 3-strand braids in the first symmetric representation, utilizing symmetry and comparison with known knot invariants.

Findings

Derived a general formula for m=3 braid extended HOMFLY polynomials.

Used symmetry to relate symmetric and antisymmetric representations.

Applied results to the figure eight knot and extended to superpolynomials.

Abstract

We continue the program of systematic study of extended HOMFLY polynomials. Extended polynomials depend on infinitely many time variables, are close relatives of integrable tau-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(\infty). Being restricted to "the topological locus" in the space of time variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m=3, when the braid is parameterized by a sequence of integers (a_1,b_1,a_2,b_2,...), and for the first non-fundamental representation R=[2]. Instead of calculating the mixing matrices directly, we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1,1]. The result applies, in particular, to the figure eight knot 4_1, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in arXiv:1203.5978.