Geodesic
Graphs on surfaces and Khovanov homology
Champanerkar, Abhijit1 ; Kofman, Ilya2 ; Stoltzfus, Neal3
1Department of Mathematics and Statistics, University of South Alabama, Mobile AL 36688, USA
2Department of Mathematics, College of Staten Island, City University of New York, 2800 Victory Boulevard, Staten Island NY 10314, USA
3Department of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918, USA
Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1531-1540
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2007.7.1531
Keywords: ribbon graphs, dessin d'enfants, quasi-trees, Khovanov homology, chord diagrams

Champanerkar, Abhijit  1   ; Kofman, Ilya  2   ; Stoltzfus, Neal  3

1 Department of Mathematics and Statistics, University of South Alabama, Mobile AL 36688, USA
2 Department of Mathematics, College of Staten Island, City University of New York, 2800 Victory Boulevard, Staten Island NY 10314, USA
3 Department of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918, USA 
@article{10_2140_agt_2007_7_1531,
     author = {Champanerkar, Abhijit and Kofman, Ilya and Stoltzfus, Neal},
     title = {Graphs on surfaces and {Khovanov} homology},
     journal = {Algebraic and Geometric Topology},
     pages = {1531--1540},
     year = {2007},
     volume = {7},
     number = {3},
     doi = {10.2140/agt.2007.7.1531},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2007.7.1531/}
}
TY  - JOUR
AU  - Champanerkar, Abhijit
AU  - Kofman, Ilya
AU  - Stoltzfus, Neal
TI  - Graphs on surfaces and Khovanov homology
JO  - Algebraic and Geometric Topology
PY  - 2007
SP  - 1531
EP  - 1540
VL  - 7
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2007.7.1531/
DO  - 10.2140/agt.2007.7.1531
ID  - 10_2140_agt_2007_7_1531
ER  - 
%0 Journal Article
%A Champanerkar, Abhijit
%A Kofman, Ilya
%A Stoltzfus, Neal
%T Graphs on surfaces and Khovanov homology
%J Algebraic and Geometric Topology
%D 2007
%P 1531-1540
%V 7
%N 3
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2007.7.1531/
%R 10.2140/agt.2007.7.1531
%F 10_2140_agt_2007_7_1531
Champanerkar, Abhijit; Kofman, Ilya; Stoltzfus, Neal. Graphs on surfaces and Khovanov homology. Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1531-1540. doi: 10.2140/agt.2007.7.1531

[1] B Bollobás, O Riordan, A polynomial invariant of graphs on orientable surfaces, Proc. London Math. Soc. $(3)$ 83 (2001) 513

[2] A Champanerkar, I Kofman, Spanning trees and Khovanov homology,

[3] O Dasbach, D Futer, E Kalfagianni, X S Lin, N Stoltzfus, Alternating sum formulae for the determinant and other link invariants,

[4] O Dasbach, D Futer, E Kalfagianni, X S Lin, N Stoltzfus, The Jones polynomial and graphs on surfaces,

[5] L H Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press (1983)

[6] L H Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1989) 105

[7] V Manturov, Minimal diagrams of classical and virtual links,

[8] P Turner, A spectral sequence for Khovanov homology with an application to $(3,q)$–torus links,

Cité par Sources :