Graphs on surfaces and Khovanov homology

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.

8 pages, 5 figures

Cited by in corpus (8)

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• Partial duals of plane graphs, separability and the graphs of knots
• Turaev genus and alternating decompositions
• On the Seifert graphs of a link diagram and its parallels
• The rational Khovanov homology of 3-strand pretzel links
• Heegaard diagrams corresponding to Turaev surfaces
• Near extremal Khovanov homology of Turaev genus one links
• On the arc index and Turaev genus of a link