A symmetric function lift of torus link homology

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.

arXiv commentAn extended abstract of this work will appear in the proceedings of FPSAC 2022