Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line

Abstract

We introduce a family of generalized Schröder polynomials $S_τ(q,t,a)$, indexed by triangular partitions $τ$ and prove that $S_τ(q,t,a)$ agrees with the Poincaré series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_τ$ associated to $τ$. For all integers $m,n,d\geq 1$ with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$ appears as a special case. It is known that these knots are algebraic, and as a result we obtain a proof of the $q=1$ specialization of the Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our Schröder polynomial computes the hook components in the Schur expansion of the symmetric function appearing in the shuffle theorem under any line, thus proving a triangular version of the $(q,t)$-Schröder theorem.