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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The $ \mathrm{SL}_3$ colored Jones polynomial of the trefoil


Authors: Stavros Garoufalidis, Hugh Morton and Thao Vuong
Journal: Proc. Amer. Math. Soc. 141 (2013), 2209-2220
MSC (2010): Primary 57N10; Secondary 57M25
Published electronically: February 4, 2013
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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 $ \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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11582-0
PII: S 0002-9939(2013)11582-0
Keywords: Colored Jones polynomial, knots, trefoil, torus knots, plethysm, rank 2 Lie algebras, Degree Conjecture, Witten-Reshetikhin-Turaev invariants
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.