THE SL<sub>3</sub> COLORED JONES POLYNOMIAL OF THE TREFOIL



Garoufalidis, Stavros, Morton, Hugh ORCID: 0000-0002-8524-2695 and Thao, Vuong
(2013) THE SL<sub>3</sub> COLORED JONES POLYNOMIAL OF THE TREFOIL. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 141 (6). pp. 2209-2220.

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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.

Item Type: Article
Additional Information: 12 pages and 1 figure
Uncontrolled Keywords: Colored Jones polynomial, knots, trefoil, torus knots, plethysm, rank 2 Lie algebras, Degree Conjecture, Witten-Reshetikhin-Turaev invariants
Depositing User: Symplectic Admin
Date Deposited: 21 Apr 2016 10:04
Last Modified: 18 Oct 2023 14:04
DOI: 10.1090/S0002-9939-2013-11582-0
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3000519