THE HOMFLYPT SKEIN ALGEBRA OF THE TORUS AND THE ELLIPTIC HALL ALGEBRA

Morton, Hugh ORCID: 0000-0002-8524-2695 and Samuelson, Peter (2017) THE HOMFLYPT SKEIN ALGEBRA OF THE TORUS AND THE ELLIPTIC HALL ALGEBRA. DUKE MATHEMATICAL JOURNAL, 166 (5). pp. 801-854.

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Abstract

We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^2$, 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 $t=q$ specialization of the elliptic Hall algebra of Burban and Schiffmann [BS12]. 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 $\lambda$-colored Homflypt polynomial of $K$. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko [CD14] using the $\mathfrak{gl}_N$ double affine Hecke algebra. This proves one of the Connection Conjectures in [CD14].

Item Type:	Article
Additional Information:	v1: preliminary version, 36 pages, many figures. v2: minor edits and improvements to exposition
Uncontrolled Keywords:	math.QA, math.QA, math.RT
Depositing User:	Symplectic Admin
Date Deposited:	14 Jul 2016 07:55
Last Modified:	19 Jan 2023 07:33
DOI:	10.1215/00127094-3718881
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3002330

Available Versions of this Item

• The Homflypt skein algebra of the torus and the elliptic Hall algebra. (deposited 22 Jun 2015 09:50)
  • The Homflypt skein algebra of the torus and the elliptic Hall algebra. (deposited 30 Jun 2016 14:01)
    • THE HOMFLYPT SKEIN ALGEBRA OF THE TORUS AND THE ELLIPTIC HALL ALGEBRA. (deposited 14 Jul 2016 07:55) [Currently Displayed]