2024/05/16 by Marko Stošić, Stošić, Marko, Piotr Sułkowski +1
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Rough Sets and Fuzzy Logic #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2405.10161
openalex publication_date 2024/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schröder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomials of this knot. The generators of uncolored HOMFLY-PT homology correspond to a basic set of such paths. Invoking the knots-quivers correspondence, we express generating functions of such paths as quiver generating series, and also relate them to quadruply-graded knot homology. Furthermore, we determine corresponding A-polynomials, which provide algebraic equations and recursion relations for generating functions of generalized Schröder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions.