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 , there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring . This correspondence preserves the bigrading used for the spanning tree model of Khovanov homology, whose Euler characteristic is the Jones polynomial of . Thus, Khovanov homology can be expressed in terms of ribbon graphs, with generators given by ordered chord diagrams.
"Graphs on surfaces and Khovanov homology." Algebr. Geom. Topol. 7 (3) 1531 - 1540, 2007. https://doi.org/10.2140/agt.2007.7.1531