Torus knots and the rational DAHA
type: Item
instance of: scholarly article
title: Torus knots and the rational DAHA
zbMATH Open document ID: 1318.57010
DOI: 10.1215/00127094-2827126
author: E. A. Gorsky
author: Alexei Oblomkov
author: Jacob Rasmussen
author: Vivek Shende
published in: Duke Mathematical Journal
publication date: 2014-12-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.4523
review text: The paper under review makes some conjectures relating the fundamental representation of rational DAHA (doubly-affine Hecke algebras or Cherednik algebras) to quantum invariants of torus knots and to Hilbert schemes of plane singularities. The reviewer finds the paper to be well-written and compelling. Given a knot \(K\) and an integer \(N\), \textit{M. Khovanov} and \textit{L. Rozansky} constructed a doubly-graded vector space \(\mathcal{H}_{N,K}\) (the \(\mathfrak{sl}_N\) homology) categorifying the quantum \(\mathfrak{sl}_N\) invariant of \(K\) (in the fundamental representation) and a triply graded vector space \(\mathcal{H}_K\) (the HOMFLY homology) which recovers the HOMFLY polynomial of \(K\) [Fundam. Math. 199, No. 1, 1--91 (2008; Zbl 1145.57009), Geom. Topol. 12, No. 3, 1387--1425 (2008; Zbl 1146.57018)]. Conjecture 1.2 gives a formula for the HOMFLY homology \(\mathcal{H}_{T(m,n)}\) of the \((m,n)\)-torus knot, identifying it with a triply graded group \(H_{m/n}\) associated to the fundamental representation of a rational DAHA. Of its three gradings, one grading comes from the degree in the exterior algebra, one from an element of the rational Cherednik algebra, and one from a certain filtration \(\mathcal{F}\) for which the paper gives several interpretations. The \(q\)-graded character of \(H_{m/n}\) is shown to coincide with the HOMFLY polynomial of the \((m, n)\)-torus knot (Proposition 1.1). Various structural properties of the HOMFLY homology of \((m,n)\)-torus knots were conjectured in [\textit{N. M. Dunfield} et al., Exp. Math. 15, No. 2, 129--159 (2006; Zbl 1118.57012)]. The paper tests Conjecture 1.2 by verifying that these indeed hold for the triply graded groups \(H_{m/n}\). In particular, \(H_{m/n}\) admits mutually anticommuting differentials which behave in the expected way and which conjecturally recover the \(\mathfrak{sl}_N\) homology from the HOMFLY homology in the expected manner (Conjecture 1.6). In the limit \(m\to \infty\) the repres…
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 16G99
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 6379682
zbMATH Keywords: Khovanov homology
zbMATH Keywords: HOMFLY polynomial
zbMATH Keywords: rational DAHA
zbMATH Keywords: rational Cherednik algebra
zbMATH Keywords: torus knot
zbMATH Keywords: Hilbert scheme
MaRDI profile type: Publication
OpenAlex ID: W3098974313
arXiv ID: 1207.4523
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description: scientific article