1 April 2017 The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
Hugh Morton, Peter Samuelson
Duke Math. J. 166(5): 801-854 (1 April 2017). DOI: 10.1215/00127094-3718881


We give a generators and relations presentation of the HOMFLYPT skein algebra H of the torus T2, and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that H is isomorphic to the σ=σ¯1 specialization of the elliptic Hall algebra of Burban and Schiffmann.

As an application, for an iterated cable K of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the λ-colored HOMFLYPT polynomial of K. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the glN double affine Hecke algebra. This proves one of the connection conjectures in their recent work.


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Hugh Morton. Peter Samuelson. "The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra." Duke Math. J. 166 (5) 801 - 854, 1 April 2017. https://doi.org/10.1215/00127094-3718881


Received: 15 May 2015; Revised: 16 May 2016; Published: 1 April 2017
First available in Project Euclid: 9 December 2016

zbMATH: 1369.16034
MathSciNet: MR3626565
Digital Object Identifier: 10.1215/00127094-3718881

Primary: 16T99
Secondary: 57M27 , 81R10

Keywords: double affine Hecke algebras , elliptic Hall algebra , HOMFLYPT polynomial , knot theory

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 5 • 1 April 2017
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