Riemann Hypothesis for DAHA superpolynomials and plane curve singularities

Stable Khovanov–Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse–Weil zeta-functions. We conjecture that all $a$-coefficients of the DAHA superpolynomials upon the substitution $q \mapsto qt$ satisfy the Riemann Hypothesis for sufficiently small $q$ for uncolored algebraic knots, presumably for $q \leq 1/2$ as $a = 0$. This can be partially extended to algebraic links at least for $a = 0$. Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov’s motivic zeta and the Galkin–Stöhr zeta-functions are discussed.

Keywords

double affine Hecke algebras, Jones polynomials, HOMFLY-PT polynomials, plane curve singularities, compactified Jacobians, Hilbert scheme, Khovanov-Rozansky homology, iterated torus links, Macdonald polynomial, Hasse-Weil zeta-function, Riemann hypothesis

2010 Mathematics Subject Classification

14H50, 17B22, 17B45, 20C08, 20F36, 30F10, 33D52, 55N10, 57M25

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The author was partially supported by NSF grant DMS-1363138.

Received 1 November 2017

Accepted 8 May 2018

Published 25 September 2018