Séminaire Lotharingien de Combinatoire, 91B.74 (2024), 12 pp.
Carmen Caprau, Nicolle González, Matthew Hogancamp and Mikhail Mazin
Triangular (q, t)-Schröder Polynomials and Khovanov-Rozansky Homology
Abstract.
We define generalized Schröder
polynomials Sλ(q,t,a) for
triangular partitions and prove that these polynomials recover the
triangular (q,t)-Catalan polynomials of [Forum
Math. Pi 11 (2023), e5, 38 pp.] at
a=0. Moreover, we show that the Poincaré polynomials of the
reduced Khovanov-Rozansky homology of Coxeter knots of these
partitions are given
by Sλ(q,t,a). Finally, combined with
recent results in [preprint,
arχiv:2210.12569],
we compute the Poincaré polynomial of
the (d,dnm+1)-cable of the (n,m)-torus knot, thus proving a
special case of the Oblomkov-Rassmusen-Shende conjecture
for generic unibranched planar curves with two Puiseux pairs.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
The following versions are available: