Khovanov-Rozansky homology and higher Catalan sequences

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 $n,m\ge 1$. 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.

arXiv comment39 pages, many figures

References in corpus (4)

• The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
• Science Fiction and Macdonald's Polynomials
• Categorified Young symmetrizers and stable homology of torus links
• q,t-Catalan numbers and knot homology

Cited by in corpus (13)

• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link
• Homology of torus knots
• Hilbert schemes and $y$-ification of Khovanov-Rozansky homology
• Khovanov polynomials for satellites and asymptotic adjoint polynomials
• Soergel bimodules and matrix factorizations
• Tautological classes and symmetry in Khovanov-Rozansky homology
• Colored Khovanov-Rozansky homology for infinite braids
• An Algebra Structure for the stable Khovanov homology of torus links
• Curved Rickard complexes and link homologies
• A categorification of the colored Jones polynomial at a root of unity
• Homology of Hilbert schemes of reducible locally planar curves
• Ext-enhanced monoidal Koszul duality for ${\mathrm{GL}}_2$
• Algebra and geometry of link homology