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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


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

ribbon graphs, dessin d'enfants, quasi-trees, Khovanov homology, chord diagrams
Mathematical Subject Classification 2000
Primary: 57M25, 57M15
Secondary: 05C10
Received: 27 May 2007
Accepted: 24 July 2007
Published: 14 November 2007
Abhijit Champanerkar
Department of Mathematics and Statistics
University of South Alabama
Mobile AL 36688
Ilya Kofman
Department of Mathematics
College of Staten Island
City University of New York
2800 Victory Boulevard
Staten Island NY 10314
Neal Stoltzfus
Department of Mathematics
Louisiana State University
Baton Rouge LA 70803-4918