A symmetric function lift of torus link homology

TL;DR

This paper introduces a new symmetric function $L_{M,N}$ that generalizes existing homology recursions for torus links, providing a novel algebraic framework and conjecturing a deep connection with elliptic Hall algebra operators.

Contribution

It defines a symmetric function $L_{M,N}$ that satisfies a generalized recursion for torus link homology and links it to elliptic Hall algebra operators, advancing the algebraic understanding of link invariants.

Findings

$L_{M,N}$ satisfies a generalized recursion for torus link homology.

$L_{M,N}$ specializes to the triply-graded Khovanov--Rozansky homology.

Conjecture relating $L_{M,N}$ to elliptic Hall algebra operators.

Abstract

Suppose $M$ and $N$ are positive integers and let $k = \gcd(M, N)$, $m = M/k$, and $n=N/k$. We define a symmetric function $L_{M,N}$ as a weighted sum over certain tuples of lattice paths. We show that $L_{M,N}$ satisfies a generalization of Mellit and Hogancamp's recursion for the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link. As a corollary, we obtain the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link as a specialization of $L_{M,N}$. We conjecture that $L_{M,N}$ is equal (up to a constant) to the elliptic Hall algebra operator $\mathbf{Q}_{m,n}$ composed $k$ times and applied to 1.