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