A categorification of Kauffman states for planar graphs

Given a decorated planar graph $(G,\omega )$, where $G$ is a planar graph and $\omega \in H^1(\mathrm{\mid }\mathcal{Q}G\mathrm{\mid },\mathbb{Z})$ 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.