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The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. (English) Zbl 1388.14087
The authors formulate a conjecture relating the Poincaré polynomial \(\overline{\mathcal P}(q,a,t)\) of the triply graded (unreduced) Khovanov-Rozansky HOMFLY homologies of an algebraic link to an other series involving the weight polynomials of certain Hilbert schemes of points supported on the corresponding complex plane curve singularity. Denoting the latter series by \(\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\), the conjectured identity \(\overline{\mathcal P}(q,a,t) = \overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\) setting \(t = -1\) specializes to an earlier conjecture of A. Oblomkov and V. Shende [Duke Math. J. 161, No. 7, 1277–1303 (2012; Zbl 1256.14025)] relating the HOMFLY polynomial of an algebraic link to a series with coefficients being the Euler characteristics of the above Hilbert scheme spaces associated to the corresponding plane curve singularity.
Several results are proved to support the proposed conjecture. The first is a statement about the structure of \(\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\) in terms of the number of branches, the arithmetic genus and the multiplicity of the singularity. A symmetry property with respect to the variable change \(t \mapsto 1/qt\) is also proved.
It is also discussed that the lowest \(a\)-degree part \(\overline{\mathcal P}_{\mathrm{alg}}^{\min}\) of \(\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\) can be computed from the compactified Jacobian of the singularity.
Next, some explicit formulas are proved for \(\overline{\mathcal P}_{\mathrm{alg}}\) in the case of algebraic torus knots, through a stratification of the Hilbert scheme type moduli space corresponding to these singularities. The limiting case \((n,\infty)\) of the \((n,k)\) torus knot results matches the formula for the stable superpolynomial of torus knots conjectured in [N. M. Dunfield et al., Exp. Math. 15, No. 2, 129–159 (2006; Zbl 1118.57012)].
In the \((n, mn+1)\) torus knot case, a further conjectural formula (based on a conjecturally identical stratification) is proposed using Cherednik algebras expressing the lowest \(a\)-degree part \(\overline{\mathcal P}_{\mathrm{alg}}^{\min}\) (and, as explained further in [E. Gorsky et al., Duke Math. J. 163, No. 14, 2709–2794 (2014; Zbl 1318.57010)], even the entire \(\overline{\mathcal P}_{\mathrm{alg}}\)) in terms of an expression involving partitions of \(n\).
The Appendix by E. Gorsky relates the Hilbert scheme stratification of torus knot type plane curve singularities to the combinatorics of diagonal harmonics and DAHA representations, justifying some conjectures involving \((q,t)\)-Catalan numbers from E. A. Gorsky [Contemp. Math. 566, 213–232 (2012; Zbl 1294.57007)].

MSC:
14H20 Singularities of curves, local rings
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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