Algebraic & Geometric Topology Volume 7 (2007)

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

Primary: 57M15, 57M25

Secondary: 05C10

References

Publication

Received: 27 May 2007
Accepted: 24 July 2007
Published: 14 November 2007

Authors

Abhijit Champanerkar
Department of Mathematics and Statistics University of South Alabama Mobile AL 36688 USA
{http://www.southalabama.edu/mathstat/personal_pages/achamp/}
Ilya Kofman
Department of Mathematics College of Staten Island City University of New York 2800 Victory Boulevard Staten Island NY 10314 USA
{http://www.math.csi.cuny.edu/~ikofman/}
Neal Stoltzfus
Department of Mathematics Louisiana State University Baton Rouge LA 70803-4918 USA
{http://www.math.lsu.edu/~stoltz/}