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Graphs on surfaces and
Khovanov homology
Abhijit Champanerkar, Ilya Kofman and Neal Stoltzfus |
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Algebraic & Geometric Topology 7 (2007) 1531–1540
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arXiv: math.GT/0705.3453 |
Abstract |
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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. |
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In memory of Xiao-Song Lin |
Keywords
ribbon graphs, dessin d'enfants, quasi-trees, Khovanov
homology, chord diagrams
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Mathematical Subject Classification 2000
Primary: 57M25, 57M15
Secondary: 05C10
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References |
Publication
Received: 27 May 2007
Accepted: 24 July 2007
Published: 14 November 2007
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Authors |
| Abhijit Champanerkar | |
| Department of Mathematics and
Statistics University of South Alabama Mobile AL 36688 USA |
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| {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 |
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| {http://www.math.csi.cuny.edu/~ikofman/} | |
| Neal Stoltzfus | |
| Department of Mathematics Louisiana State University Baton Rouge LA 70803-4918 USA |
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| {http://www.math.lsu.edu/~stoltz/} | |
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