Knot polynomials in the first non-symmetric representation

TL;DR

This paper derives the explicit HOMFLY polynomial for the figure eight knot and other 3-strand knots in a non-symmetric representation, revealing new structural insights and computational techniques beyond symmetric cases.

Contribution

It presents the first explicit results for non-torus knots in non-symmetric representations and introduces methods for their calculation and structural understanding.

Findings

Explicit HOMFLY polynomial for figure eight knot in [21] representation

Development of computational techniques for non-symmetric representations

Discussion of superpolynomial formulas and conjectural expansions

Abstract

We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure eight and some other 3-strand knots in representation [21]. This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole variety of different arguments, allowing one to guess the answer for the figure eight knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure eight and some other 3-strand knots based on the previously developed sophisticated technique of multi-strand calculations. We also discuss a formula for the superpolynomial in representation [21] for the figure eight knot, which heavily relies on the conjectural form of superpolynomial expansion nearby the special polynomial point. Generalizations and details will be presented elsewhere.