Generalised Kauffman Clock Theorems

TL;DR

This work extends Kauffman’s Clock Theorem from planar 4-valent graphs to graphs embedded on arbitrary compact oriented surfaces by introducing multiverses, spines, and framings. The authors prove two clock theorems: a planar version yielding a distributive lattice of states with plane transpositions as covers, and a genus-general version that fixes a viable circulation and uses surface transpositions to obtain componentwise distributive lattices. The analysis hinges on Propp’s lattice theory for matchings/orientations and a spine duality framework that translates states to matchings and orientations, enabling a unified lattice-theoretic treatment across genera. The results recover Kauffman’s clock lattice for universes on discs and establish new lattice structures for multiverses in positive genus, with direct connections to planar overlaid Tait graphs and to the broader combinatorics of matchings and orientations on embedded graphs.

Abstract

Kauffman's clock theorem provides a distributive lattice structure on the set of states of a four-valent graph in the plane. We prove two distinct generalisations of this theorem, for four-valent graphs embedded in more general compact oriented surfaces. The proofs use results of Propp providing distributive lattice structures on matchings on bipartite plane graphs, and orientations of graphs with fixed circulation.

Figures (24)

• Figure 1: Top left and right: multiverses on a disc. Bottom: two depictions of the same multiverse on a punctured torus.
• Figure 2: The Hasse diagram of the lattice obtained from Theorem \ref{['Thm:main_thm_1']}, for a framed planar multiverse.
• Figure 3: The Hasse diagram of the lattice obtained from Theorem \ref{['Thm:main_thm_2']}, for a framed planar multiverse.
• Figure 4: The Hasse diagram of the lattice obtained from Kauffman's clock theorem or Theorem \ref{['Thm:main_thm_1']}, for a Kauffman string universe.
• Figure 5: The Hasse diagram of the lattice obtained from Theorem \ref{['Thm:main_thm_2']}, for a Kauffman string universe.

Theorems & Definitions (177)

• Theorem 1.1: Planar Clock Theorem
• Theorem 1.2: Arbitrary Genus Clock Theorem
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Theorem 1.8: Kauffman's Clock Theorem
• Definition 1.9: Universe on a disc
• Theorem 1.10: Kauffman's Clock Theorem for universes on a disc