Torus knots and the rational DAHA
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 representation theory of rational DAHA simplifies considerably, and Proposition 1.7 gives an explicit formula is given for its differential. This leads to a conjecture for the stable reduced \(\mathfrak{sl}_N\) homology of \(T(n,\infty)\) (Conjecture 1.8). The paper also explains the relation of Conjecture 1.2 to a conjecture of the first author relating knot homology to \((q,t)\)-Catalan numbers [\textit{E. A. Gorsky}, Contemp. Math. 566, 213--232 (2012; Zbl 1294.57007)] and to a conjecture of the last three authors relating unreduced HOMFLY homology to cohomologies of certain nested Hilbert schemes of points [\textit{A. Oblomkov}, \textit{J. Rasmussen} and \textit{V. Shende}, The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link, preprint, \url{arXiv:1201.2115}].
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