Work

Torus knots and generalized Schröder paths

Year: 2025

Type: article

Abstract: 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.

Source: Nuclear Physics B

DOI: 10.1016/j.nuclphysb.2025.116814

Authors: Marko Stošić, Piotr Sułkowski

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Narodowe Centrum Nauki, Narodowym Centrum Nauki, Science Fund of the Republic of Serbia, Fundação para a Ciência e a Tecnologia

Awards: G2007980322, Knots, quivers and categorifications, G4118941449, G6997658380, G8718594641

OpenAlex ID: W4407099628

FWCI:

Cites: 27

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Related to: 10