The SL_3 colored Jones polynomial of the trefoil

TL;DR

This paper derives an explicit formula for the $sl_3$ colored Jones polynomial of T(2,n) torus knots, including the trefoil, by computing the second plethysm of $sl_3$ representations, enabling new verifications and conjectures.

Contribution

It provides the first explicit formula for the second plethysm of any $sl_3$ representation, leading to a closed-form expression for the $sl_3$ colored Jones polynomial of T(2,n) knots.

Findings

Verified the Degree Conjecture for T(2,n) torus knots.

Efficiently computed $sl_3$ Witten-Reshetikhin-Turaev invariants of the Poincaré sphere.

Guessed a Groebner basis for the recursion ideal of the $sl_3$ colored Jones polynomial of the trefoil.

Abstract

Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, which allows us to give an explicit formula for the colored Jones polynomial of the trefoil, and more generally, for T(2,n) torus knots. We give two independent proofs of our plethysm formula, one of which uses the work of Carini-Remmel. Our formula for the $\fsl_3$ colored Jones polynomial of T(2,n) torus knots allows us to verify the Degree Conjecture for those knots, to efficiently the $\fsl_3$ Witten-Reshetikhin-Turaev invariants of the Poincare sphere, and to guess a Groebner basis for recursion ideal of the $\fsl_3$ colored Jones polynomial of the trefoil.