Exceptional knot homology

TL;DR

This paper introduces hyperpolynomials as new invariants for torus knots, linking DAHA to BPS states and addressing negative coefficient issues in homological knot invariants.

Contribution

It establishes a natural connection between double affine Hecke algebras and BPS states, and introduces hyperpolynomials to improve homological invariants of torus knots.

Findings

Hyperpolynomials effectively address negative coefficient problems.

A new structure of differentials on homological invariants is described.

Connections between physics, singularity spectra, and knot invariants are established.

Abstract

The goal of this article is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called "hyperpolynomials" that address the "problem of negative coefficients" often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau-Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.