Title: 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 $n,m\geq 1$. 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.