Schr\"oder Paths, Their Generalizations and Knot Invariants

TL;DR

This paper explores generalized Schr"oder paths with rational slopes, deriving their generating functions and linking them to knot invariants like the HOMFLY-PT polynomials and superpolynomials of torus knots.

Contribution

It introduces new generalizations of Schr"oder paths, establishes their $q$-difference equations, and connects these combinatorial objects to knot invariants, providing combinatorial proofs of recent results.

Findings

Derived $q$-difference equations for generating functions

Established relations between Schr"oder paths and HOMFLY-PT polynomials

Provided combinatorial proof linking paths to knot homology

Abstract

We study some kinds of generalizations of Schr\"oder paths below a line with rational slope and derive the $q$-difference equations that are satisfied by their generating functions. As a result, we establish a relation between the generating function of generalized Schr\"oder paths with backwards and the wave function corresponding to colored HOMFLY-PT polynomials of torus knot $T_{1,f}$. We also give a combinatorial proof of a recent result by Sto\v{s}i\'c and Su{\l}kowski, in which the standard generalized Schr\"oder paths are related to the superpolynomial of reduced colored HOMFLY-PT homology of $T_{1,f}$.