Iterated Torus Knots and Double Affine Hecke Algebras. (4th September 2017)

• Record Type: Journal Article
• Authors: Samuelson, Peter
• Is Part Of: International mathematics research notices. Volume 2019:Number 9(2019)
• Page Start: 2848
• Page End: 2893
• Publication Date: 2017-09-04
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx198
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We give a topological realization of the (spherical) double affine Hecke algebra $\mathrm{SH}_{q, t}$ of type ${\mathfrak{sl}}_2$, and we use this to construct a module over $\mathrm{SH}_{q, t}$ for any knot $K \subset S^3$ . As an application, we give a purely topological interpretation of Cherednik's two-variable polynomials $P_n(r, s; q, t)$ of type ${\mathfrak{sl}}_2$ from [14 ] (where $r, s \in {\mathbb{Z}}$ are relatively prime). We then generalize the construction of these polynomials (for ${\mathfrak{sl}}_2$ ) from torus knots to all iterated cables of the unknot and prove they specialize to the colored Jones polynomials of the knot. Finally, in the Appendix we compare this construction to a later construction of Cherednik and Danilenko.