Khovanov-Rozansky homology of Coxeter knots and Schr\"oder polynomials for paths under any line

TL;DR

This paper introduces generalized Schr"oder polynomials linked to Khovanov-Rozansky homology of Coxeter knots, proving conjectures and connecting to symmetric functions and the shuffle theorem.

Contribution

It establishes a new family of polynomials that match the Poincaré series of triply graded homology for Coxeter knots, linking knot invariants to symmetric functions and the shuffle theorem.

Findings

Generalized Schr"oder polynomials match Khovanov-Rozansky homology Poincaré series.

Proof of the $q=1$ case of the Oblomkov-Rasmussen-Shende conjecture for algebraic knots.

Schr"oder polynomials compute hook components in Schur expansions related to the shuffle theorem.

Abstract

We introduce a family of generalized Schr\"oder polynomials $S_\tau(q,t,a)$, indexed by triangular partitions $\tau$ and prove that $S_\tau(q,t,a)$ agrees with the Poincar\'e series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_\tau$ associated to $\tau$. 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 Schr\"oder 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 $(q,t)$-Schr\"oder theorem.