Khovanov-Rozansky homology and higher Catalan sequences

Matthew Hogancamp

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arXiv · 1704.01562

Khovanov-Rozansky homology and higher Catalan sequences

Matthew Hogancamp

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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.

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BibTeX RIS

Matthew Hogancamp. 2017-04-05. Khovanov-Rozansky homology and higher Catalan sequences. https://arxiv.org/abs/1704.01562

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