1. Schr\'oder Paths, Their Generalizations and Knot Invariants
- Author
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Ji, Ce, Tang, Qian, and Yang, Chenglang
- Subjects
Mathematics - Combinatorics ,Mathematical Physics - 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 $T_{1,f}$., Comment: 16 pages
- Published
- 2024