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Double affine Hecke algebras and generalized Jones polynomials

Part of: Representation theory of groups Low-dimensional topology

Published online by Cambridge University Press:  01 April 2016

Yuri Berest  and
Peter Samuelson
Yuri Berest
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA email [email protected]
Peter Samuelson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M4Y 1H5, Canada email [email protected] Current address: Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
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Abstract

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

Keywords

double affine Hecke algebrasknotsskein modulesquantum torusKauffman bracketJones polynomialcharacter variety

MSC classification

Primary: 57M25: Knots and links in $S^3$ 20C08: Hecke algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 7 , July 2016 , pp. 1333 - 1384
Copyright
© The Authors 2016 

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