Torus knots and generalized Schr\"oder paths

TL;DR

This paper establishes a deep connection between torus knot invariants and generalized Schr"oder paths, providing new generating functions, algebraic equations, and relations to knot homologies and BPS states.

Contribution

It introduces generalized Schr"oder paths to encode torus knot invariants and relates these paths to HOMFLY-PT polynomials, quiver series, and algebraic A-polynomials.

Findings

Generated functions encode colored HOMFLY-PT polynomials.

Established relations between paths and knot homologies.

Derived A-polynomials as algebraic equations for paths.

Abstract

We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schr\"oder 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 Schr\"oder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions.