Volume 7, issue 3 (2007)

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Graphs on surfaces and Khovanov homology

Abhijit Champanerkar, Ilya Kofman and Neal Stoltzfus

Algebraic & Geometric Topology 7 (2007) 1531–1540
 arXiv: math.GT/0705.3453
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.

 In memory of Xiao-Song Lin
Keywords
ribbon graphs, dessin d'enfants, quasi-trees, Khovanov homology, chord diagrams
Mathematical Subject Classification 2000
Primary: 57M25, 57M15
Secondary: 05C10