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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
DOI: https://doi.org/10.1090/S0002-9939-2013-11582-0
Published electronically: February 4, 2013
MathSciNet review: 3034446
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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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  • [BN05] Dror Bar-Natan, Knotatlas, 2005, http://katlas.org.
  • [CGR84] Y. M. Chen, A. M. Garsia, and J. Remmel, Algorithms for plethysm, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 109-153. MR 777698 (86f:05010)
  • [CR98] Luisa Carini and J. B. Remmel, Formulas for the expansion of the plethysms $ s_2[s_{(a,b)}]$ and $ s_2[s_{(n^k)}]$, Discrete Math. 193 (1998), no. 1-3, 147-177, Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661367 (2000b:05129)
  • [FH91] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991, A first course, Readings in Mathematics. MR 1153249 (93a:20069)
  • [GK10] Stavros Garoufalidis and Christoph Koutschan, The $ \mathfrak{sl}_3$ Jones polynomial of the trefoil: a case study of $ q$-holonomic sequences, Advances in Applied Math. 47 (4) (2011), 829-839. MR 2832380
  • [GV] Stavros Garoufalidis and Thao Vuong, The degree conjecture for torus knots, preprint, 2010.
  • [Jan96] Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532 (96m:17029)
  • [Jon87] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335-388. MR 908150 (89c:46092)
  • [Law03] Ruth Lawrence, The $ \rm PSU(3)$ invariant of the Poincaré homology sphere, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop ``Invariants of Three-Manifolds'' (Calgary, AB, 1999), Topology Appl. 127 (2003), 153-168. MR 1953324 (2003m:57032)
  • [Mac95] I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, with contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144 (96h:05207)
  • [MM08] H. R. Morton and P. M. G. Manchón, Geometrical relations and plethysms in the Homfly skein of the annulus, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 305-328. MR 2439627 (2009h:57021)
  • [Mor95] H. R. Morton, The coloured Jones function and Alexander polynomial for torus knots, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 1, 129-135. MR 1297899 (95h:57008)
  • [RJ93] Marc Rosso and Vaughan Jones, On the invariants of torus knots derived from quantum groups, J. Knot Theory Ramifications 2 (1993), no. 1, 97-112. MR 1209320 (94a:57019)
  • [Tur88] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527-553. MR 939474 (89e:57003)
  • [Tur94] -, Quantum invariants of knots and $ 3$-manifolds, de Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673 (95k:57014)

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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: https://doi.org/10.1090/S0002-9939-2013-11582-0
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.

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