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arXiv · 2407.20010

Schr\"oder Paths, Their Generalizations and Knot Invariants

Ce Ji·Qian Tang·Chenglang Yang
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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 $T_{1,f}$.

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Ce Ji, Qian Tang, Chenglang Yang. 2024-07-29. Schr\"oder Paths, Their Generalizations and Knot Invariants. https://arxiv.org/abs/2407.20010

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