Tautological classes for (n,n+1) torus knots

TL;DR

This paper establishes a deep connection between HOMFLY-PT homology of (n,n+1) torus knots and diagonal coinvariants, revealing new algebraic actions and computing differential actions in spectral sequences.

Contribution

It constructs an explicit isomorphism linking HOMFLY-PT homology of (n,n+1) torus knots to diagonal coinvariants and extends tautological class actions to a Lie algebra of Hamiltonian vector fields.

Findings

Explicit isomorphism between HOMFLY-PT homology and diagonal coinvariants.

Extension of tautological class actions to a Lie algebra of Hamiltonian vector fields.

Computation of differentials in Rasmussen spectral sequences for these knots.

Abstract

We construct an explicit isomorphism between the HOMFLY-PT homology of $(n,n+1)$ torus knots and the direct sum of hook isotypic components of the space of diagonal coinvariants. As a consequence, we compute the action of tautological classes in HOMFLY-PT homology of $(n,n+1)$ torus knots and prove that it extends to an action of the Lie algebra of Hamiltonian vector fields on the plane. We also compute the action of differentials $d_N$ in Rasmussen spectral sequences from HOMLFY-PT to $\mathfrak{gl}(N)$ homology of $(n,n+1)$ torus knots.