The $\mathrm {SL}_3$ 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 $\mathfrak {sl}_3$, 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 and Remmel. Our formula for the $\mathfrak {sl}_3$ colored Jones polynomial of $T(2,n)$ torus knots allows us to verify the Degree Conjecture for those knots, to efficiently determine the $\mathfrak {sl}_3$ Witten-Reshetikhin-Turaev invariants of the Poincaré sphere, and to guess a Groebner basis for the recursion ideal of the $\mathfrak {sl}_3$ colored Jones polynomial of the trefoil.