Khovanov-Rozansky homology and higher Catalan sequences

2017/04/05 by Matthew Hogancamp, Hogancamp, Matthew · 6 citations
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1704.01562

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± 1) and (n,nm) torus links for n,m≥ 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.

Cited by

• Doubly-graded exponential growth of colored torus knot homology
• Tautological classes and symmetry in Khovanov-Rozansky homology
• The minimal Rickard complexes of braids on two strands
• Ext-enhanced monoidal Koszul duality for GL₂
• Homology of Hilbert schemes of reducible locally planar curves
• A categorification of the colored Jones polynomial at a root of unity

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• Torus link homology
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• Macdonald polynomials and operators and Catalan Combinatorics
• A symmetric function lift of torus link homology