Donaldson-Thomas invariants, torus knots, and lattice paths

TL;DR

This paper establishes a deep connection between quiver representations, torus knot invariants, and lattice path combinatorics, providing explicit formulas and algebraic structures linking these areas.

Contribution

It introduces explicit formulas for Donaldson-Thomas invariants of symmetric quivers and relates them to lattice path counting and knot invariants, including quantum generalizations.

Findings

Explicit formulas for classical generating functions and invariants of symmetric quivers.

Correspondence between extremal quiver generating functions and lattice path counting under a line.

Quantum extensions linking motivic quiver functions, quantum knot invariants, and q-weighted paths.

Abstract

In this paper we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory -- we find explicit formulae for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to $(r,s)$ torus knots and show that their classical generating functions, in the extremal limit and framing $rs$, are generating functions of lattice paths under the line of the slope $r/s$. Generating functions of such paths satisfy extremal A-polynomial equations, which immediately follows after representing them in terms of the Duchon grammar. Moreover, these extremal A-polynomial equations encode Donaldson-Thomas invariants, which provides an interesting example of algebraicity of generating functions of these invariants. We also find a quantum generalization of these statements, i.e. a relation between motivic quiver generating functions, quantum extremal knot invariants, and $q$-weighted path counting. Finally, in the case of the unknot, we generalize this correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder paths.