The colored Jones polynomial of the trefoil
Authors:
Stavros Garoufalidis, Hugh Morton and Thao Vuong
Journal:
Proc. Amer. Math. Soc. 141 (2013), 22092220
MSC (2010):
Primary 57N10; Secondary 57M25
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 RossoJones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of , which allows us to give an explicit formula for the colored Jones polynomial of the trefoil and, more generally, for 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 colored Jones polynomial of torus knots allows us to verify the Degree Conjecture for those knots, to efficiently determine the WittenReshetikhinTuraev invariants of the Poincaré sphere, and to guess a Groebner basis for the recursion ideal of the colored Jones polynomial of the trefoil.
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Remmel, Algorithms for plethysm, Combinatorics and algebra
(Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc.,
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 Dror BarNatan, 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. 109153. MR 777698 (86f:05010)
 [CR98]
 Luisa Carini and J. B. Remmel, Formulas for the expansion of the plethysms and , Discrete Math. 193 (1998), no. 13, 147177, 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, SpringerVerlag, New York, 1991, A first course, Readings in Mathematics. MR 1153249 (93a:20069)
 [GK10]
 Stavros Garoufalidis and Christoph Koutschan, The Jones polynomial of the trefoil: a case study of holonomic sequences, Advances in Applied Math. 47 (4) (2011), 829839. 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, 335388. MR 908150 (89c:46092)
 [Law03]
 Ruth Lawrence, The invariant of the Poincaré homology sphere, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop ``Invariants of ThreeManifolds'' (Calgary, AB, 1999), Topology Appl. 127 (2003), 153168. 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, 305328. 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, 129135. 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, 97112. MR 1209320 (94a:57019)
 [Tur88]
 V. G. Turaev, The YangBaxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527553. MR 939474 (89e:57003)
 [Tur94]
 , Quantum invariants of knots and 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 303320160
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 303320160
Email:
tvuong@math.gatech.edu
DOI:
http://dx.doi.org/10.1090/S000299392013115820
PII:
S 00029939(2013)115820
Keywords:
Colored Jones polynomial,
knots,
trefoil,
torus knots,
plethysm,
rank 2 Lie algebras,
Degree Conjecture,
WittenReshetikhinTuraev 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.
