Iterated torus knots and double affine Hecke algebras

Peter Samuelson

Research output: Contribution to journalArticlepeer-review

Abstract

We give a topological realization of the (spherical) double affine Hecke algebra SHq,t of type sl2, and we use this to construct a module over SHq,t for any knot KS3 . As an application, we give a purely topological interpretation of Cherednik's 2-variable polynomials Pn (r,s;q,t) of type sl2  from [Che13] (where r,s ∈ Z are relatively prime), and we give a new proof that these specialize to the colored Jones polynomials of the r,s torus knot.
We then generalize Cherednik's construction (for sl2 ) to all iterated cables of the unknot and prove the corresponding specialization property. Finally, in the appendix we compare our polynomials associated to iterated torus knots to the ones recently defined in [CD14], in the specialization t=−q 2 .
Original languageEnglish
Pages (from-to) 2848–2893
Number of pages28
JournalInternational Mathematics Research Notices
Volume2019
Issue number9
DOIs
Publication statusPublished - 4 Sep 2017

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