A note on colored HOMFLY polynomials for hyperbolic knots from WZW models

TL;DR

This paper develops methods to compute colored HOMFLY polynomials for hyperbolic knots using WZW models, extending previous work by including non-trivial multiplicities and providing explicit calculations for complex representations.

Contribution

It introduces a novel approach to calculate colored HOMFLY invariants for hyperbolic knots via crossing matrices and quantum 6j-symbols, covering more general representations.

Findings

Derived colored HOMFLY polynomials for two-bridge hyperbolic knots.

Extended calculations to include non-trivial multiplicities in crossing matrices.

Provided explicit HOMFLY polynomials for knots with up to eight crossings.

Abstract

Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group U_q(su(N)). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation {2,1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or -- in the limit to classical groups -- to determine color factors for Yang Mills amplitudes.