Enumeration of chord diagrams via topological recursion and quantum curve techniques

TL;DR

This paper develops methods using topological recursion and quantum curves to systematically count orientable and non-orientable chord diagrams, linking matrix models to topological enumeration.

Contribution

It introduces a novel approach combining topological recursion and quantum curve techniques to enumerate chord diagrams via matrix models.

Findings

Efficient algorithms for counting chord diagrams with specified genus, backbones, and chords.

Connection established between matrix model expectation values and topological enumeration.

Independent validation of methods using both topological recursion and quantum curves.

Abstract

In this paper we consider the enumeration of orientable and non-orientable chord diagrams. We show that this enumeration is encoded in appropriate expectation values of the $\beta$-deformed Gaussian and RNA matrix models. We evaluate these expectation values by means of the $\beta$-deformed topological recursion, and - independently - using properties of quantum curves. We show that both these methods provide efficient and systematic algorithms for counting of chord diagrams with a given genus, number of backbones and number of chords.