Torus link homology and the nabla operator

TL;DR

This paper provides a combinatorial formula for the triply graded Hochschild homology of certain links, including the (n,n) torus link, connecting it to the nabla operator on symmetric functions.

Contribution

It introduces a new combinatorial formula for the homologies of links studied by Elias and Hogancamp, and transforms it into a computable form, also proposing a conjectural link to symmetric functions.

Findings

Derived a combinatorial formula for link homologies

Transformed the formula into a computable version

Conjectured a relationship with the nabla operator on symmetric functions

Abstract

In recent work, Elias and Hogancamp develop a recurrence for the Poincar\'e series of the triply graded Hochschild homology of certain links, one of which is the $(n,n)$ torus link. In this case, Elias and Hogancamp give a combinatorial formula for this homology that is reminiscent of the combinatorics of the modified Macdonald polynomial eigenoperator $\nabla$. We give a combinatorial formula for the homologies of all links considered by Elias and Hogancamp. Our first formula is not easily computable, so we show how to transform it into a computable version. Finally, we conjecture a direct relationship between the $(n,n)$ torus link case of our formula and the symmetric function $\nabla p_{1^n}$.