Graphs on surfaces and Khovanov homology

TL;DR

This paper establishes a connection between ribbon graphs, quasi-trees, and Khovanov homology, enabling the expression of Khovanov homology in terms of ribbon graphs and chord diagrams, linking topological invariants to graph theory.

Contribution

It introduces a bijective correspondence between quasi-trees of ribbon graphs and spanning trees of checkerboard-colored graphs, relating Khovanov homology to ribbon graph structures.

Findings

Quasi-trees correspond bijectively to spanning trees in associated graphs.

Khovanov homology can be expressed via ribbon graphs and chord diagrams.

The Euler characteristic of the spanning tree model equals the Jones polynomial.

Abstract

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram $L$, there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring $L$. This correspondence preserves the bigrading used for the spanning tree model of Khovanov homology, whose Euler characteristic is the Jones polynomial of $L$. Thus, Khovanov homology can be expressed in terms of ribbon graphs, with generators given by ordered chord diagrams.