Khovanov-Rozansky homology and higher Catalan sequences

TL;DR

This paper introduces a recursive method to compute Khovanov-Rozansky homology for certain knots and links, linking it to Catalan combinatorics and confirming several conjectures related to Hilbert schemes and rational DAHA.

Contribution

It provides a simple recursion for computing homology of infinite families of knots and links, and connects these results to Catalan combinatorics, confirming multiple conjectures.

Findings

Recursion computes homology for (n,nm±1) and (n,nm) torus links.

Results confirm Gorsky's conjecture and match predictions from Hilbert schemes.

Supports topological interpretation of symmetric functions in the m-shuffle conjecture.

Abstract

We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms of Catalan combinatorics, proving a conjecture of Gorsky's. Our computations agree with predictions coming from Hilbert schemes and rational DAHA, which also proves the Gorsky-Oblomkov-Rasmussen-Shende conjectures in these cases. Additionally, our results suggest a topological interpretation of the symmetric functions which appear in the context of the $m$-shuffle conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov.