Clock theorems for triangulated surfaces

TL;DR

This paper extends Clock Theorems to triangulated surfaces, showing how matchings form distributive lattices on spheres and tori, with different connectivity properties.

Contribution

It generalizes Clock Theorems to triangulations of surfaces, revealing lattice structures and connectivity differences between sphere and torus cases.

Findings

On the sphere, matchings form a distributive lattice after removing one triangle.

On the torus, the state transition graph is often disconnected with some components forming lattices.

The local operation about black triangles acts as a state transposition.

Abstract

We investigate triangulations of the two-dimensional sphere and torus with the faces properly colored white and black. We focus on matchings between white triangles and incident vertices. On the torus our objects are perfect pairings, whereas on the sphere this is only true after removing one triangle and its vertices. In the latter case, such matchings (first studied by Tutte) extend the notion of state in Kauffman's formal knot theory and we show that his Clock Theorem, in its form due to Gilmer and Litherland, also extends: the set of matchings naturally forms a distributive lattice. Here the role of state transposition is played by a simple local operation about black triangles. By contrast, on the torus, the analogous state transition graph is usually disconnected: some of its components still form distributive lattices with global maxima and minima, while other components contain directed cycles and are without local extrema.