## 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.## References

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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