A discrete Morse perspective on knot projections and a generalised clock theorem

TL;DR

This paper characterizes matchings on Tait graphs that induce discrete Morse functions on the sphere, generalizes key theorems in knot theory, and explores their combinatorial and topological properties.

Contribution

It provides a complete characterization of matchings inducing discrete Morse functions, extends the Clock Theorem, and analyzes the related complexes and transformations.

Findings

Characterization of matchings inducing discrete Morse functions on $S^2$

A closed formula for counting such discrete Morse functions

Establishment of bijections and move sequences relating perfect matchings

Abstract

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on $S^2$, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.