A correspondence between complexes and knots

TL;DR

This paper explores the relationship between perfect matchings in bipartite graphs derived from knot diagrams and discrete Morse functions on 2-complexes, providing insights into topological and combinatorial structures.

Contribution

It establishes a novel correspondence between combinatorial graph matchings and topological Morse functions related to knots and 2-complexes.

Findings

Perfect matchings correspond to discrete Morse functions on 2-complexes.

The work links knot diagrams to topological and combinatorial structures.

Provides a new perspective for understanding knots via graph theory.

Abstract

In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded below for the opposite audience: those who may be unfamiliar with knots.