Refined knot invariants and Hilbert schemes

TL;DR

This paper advances the understanding of refined knot invariants by proving a conjecture on superpolynomial stabilization, relating invariants to Hilbert schemes, and computing explicit cases, with implications for the shuffle conjecture.

Contribution

It proves Cherednik's conjecture on superpolynomial stabilization and connects knot invariants to Hilbert schemes and rational Cherednik algebras, providing explicit computations and new conjectures.

Findings

Proof of Cherednik's conjecture on superpolynomial stabilization.

Relation of knot invariants to Hilbert schemes of points on the plane.

Explicit computation of invariants in the uncolored case.

Abstract

We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on the plane. Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m/n shuffle conjecture.