Combinatorial Theory

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 Hogancamp and Mellit'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.

Mathematics Subject Classifications: 05E05, 57M27

Keywords: Lattice paths, Dyck paths, link homology, torus links, elliptic Hall algebra, LLT polynomials