Riemann Hypothesis for DAHA superpolynomials and plane curve singularities

TL;DR

This paper explores a conjectural link between DAHA superpolynomials, algebraic knot invariants, and the Riemann Hypothesis, proposing that certain coefficients satisfy RH-like properties for small q, with implications for algebraic links and motivic zeta-functions.

Contribution

It introduces a novel conjecture connecting DAHA superpolynomials with the Riemann Hypothesis, extending the framework to algebraic links and relating to motivic zeta-functions.

Findings

Conjecture that $a$-coefficients of DAHA superpolynomials satisfy RH for small q.

Partial extension of the conjecture to algebraic links at $a=0$.

Discussion of connections with motivic and Galkin-Stöhr zeta-functions.

Abstract

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 $q\le 1/2$ 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-St\"ohr zeta-functions are discussed.