Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.19872 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918511672033280 |
|---|---|
| author | Irelli, Giovanni Cerulli Fiorenza, Domenico Landi, Eugenio Matteucci, Michele |
| author_facet | Irelli, Giovanni Cerulli Fiorenza, Domenico Landi, Eugenio Matteucci, Michele |
| contents | Given a decorated planar graph $(G,ω)$, where $G$ is a planar graph and $ω\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $ω$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,ω)$, which is the directed graph of such functions, and $BMS(G,ω)$, the directed graph of BMS states which are some pairs of $ω$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,ω)$ 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 $\mathcal{L}(G,ω)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte--Schiffler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A categorification of Kauffman states for planar graphs Irelli, Giovanni Cerulli Fiorenza, Domenico Landi, Eugenio Matteucci, Michele Representation Theory Combinatorics General Topology 57K10, 16G20, 05C10, 06D99 Given a decorated planar graph $(G,ω)$, where $G$ is a planar graph and $ω\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $ω$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,ω)$, which is the directed graph of such functions, and $BMS(G,ω)$, the directed graph of BMS states which are some pairs of $ω$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,ω)$ 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 $\mathcal{L}(G,ω)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte--Schiffler. |
| title | A categorification of Kauffman states for planar graphs |
| topic | Representation Theory Combinatorics General Topology 57K10, 16G20, 05C10, 06D99 |
| url | https://arxiv.org/abs/2605.19872 |