The 𝑆𝐿₃ colored Jones polynomial of the trefoil

Author:

Garoufalidis Stavros,Morton Hugh,Vuong Thao

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 s l 3 \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 ) 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 s l 3 \mathfrak {sl}_3 colored Jones polynomial of T ( 2 , n ) T(2,n) torus knots allows us to verify the Degree Conjecture for those knots, to efficiently determine the s l 3 \mathfrak {sl}_3 Witten-Reshetikhin-Turaev invariants of the Poincaré sphere, and to guess a Groebner basis for the recursion ideal of the s l 3 \mathfrak {sl}_3 colored Jones polynomial of the trefoil.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference15 articles.

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5. The 𝔰𝔩₃ Jones polynomial of the trefoil: a case study of 𝔮-holonomic sequences;Garoufalidis, Stavros;Adv. in Appl. Math.,2011

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