A categorification of Kauffman states for planar graphs

TL;DR

This paper introduces a categorification framework for Kauffman states in planar graphs, establishing lattice structures and representations that generalize classical knot theory results.

Contribution

It defines new directed graphs and potential-based representations, extending Kauffman's Clock Theorem to broader classes of planar graphs and states.

Findings

$ ext{L}(G, ext{ extomega})$ forms a graded distributive lattice under certain conditions.

Established an isomorphism between $ ext{L}(G, ext{ extomega})$ and subrepresentations of a maximal quiver representation.

Generalized Bazier-Matte--Schiffler's result to a wider class of graph states.

Abstract

Given a decorated planar graph $(G,\omega)$, where $G$ is a planar graph and $\omega\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $\omega$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,\omega)$, which is the directed graph of such functions, and $BMS(G,\omega)$, the directed graph of BMS states which are some pairs of $\omega$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,\omega)$ to be a graded distributive lattice, recovering Kauffman's Clock Theorem when $G$ is a knot diagram. We also define a potential on $\mathcal{Q} G$ and associate a representation of the corresponding quiver with potential to every BMS state. Under suitable assumptions, this construction yields an isomorphism between $\mathcal{L}(G,\omega)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte--Schiffler.