Schröder Paths, Their Generalizations and Knot Invariants

2024/07/29 by Ce Ji, Qiang Tang, Ji, Ce +3
Engineering · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Physics and Engineering Research Articles

paper · pdf · doi:10.48550/arxiv.2407.20010

openalex publication_date 2024/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study some kinds of generalizations of Schröder 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 Schröder 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 Stošić and Sułkowski, in which the standard generalized Schröder paths are related to the superpolynomial of reduced colored HOMFLY-PT homology of T_1,f.

Related

• Torus knots and generalized Schröder paths
• Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line
• Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials
• Doubly-graded exponential growth of colored torus knot homology
• Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials