Schr\"oder Paths, Their Generalizations and Knot Invariants

Ce Ji; Qian Tang; Chenglang Yang

Schr\"oder Paths, Their Generalizations and Knot Invariants

Combinatorics 2024-07-30 v1 Mathematical Physics math.MP

Authors: Ce Ji , Qian Tang , Chenglang Yang

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 $T_{1,f}$ .

Keywords

graph polynomials schubert polynomials

Cite

@article{arxiv.2407.20010,
  title  = {Schr\"oder Paths, Their Generalizations and Knot Invariants},
  author = {Ce Ji and Qian Tang and Chenglang Yang},
  journal= {arXiv preprint arXiv:2407.20010},
  year   = {2024}
}

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16 pages

Schr\"oder Paths, Their Generalizations and Knot Invariants

Abstract

Keywords

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