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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The $\mathrm {SL}_3$ colored Jones polynomial of the trefoil
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by Stavros Garoufalidis, Hugh Morton and Thao Vuong PDF
Proc. Amer. Math. Soc. 141 (2013), 2209-2220 Request permission

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.
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Additional Information
  • Stavros Garoufalidis
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
  • Email: stavros@math.gatech.edu
  • Hugh Morton
  • Affiliation: Department of Mathematics, University of Liverpool, Liverpool L69 3BX, England
  • Email: morton@liverpool.ac.uk
  • Thao Vuong
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
  • Email: tvuong@math.gatech.edu
  • Received by editor(s): December 6, 2010
  • Received by editor(s) in revised form: September 30, 2011
  • Published electronically: February 4, 2013
  • Additional Notes: The first author was supported in part by NSF
  • Communicated by: Daniel Ruberman
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 2209-2220
  • MSC (2010): Primary 57N10; Secondary 57M25
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11582-0
  • MathSciNet review: 3034446