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International conference "Analysis and Singularities" dedicated to the 75th anniversary of Vladimir Igorevich Arnold
December 17, 2012 15:05–15:35, Moscow, Dorodnitsyn Computing Center of the RAS (Vavilova, 40), 3rd floor

Graphs on surfaces via planar graphs

S. V. Chmutov
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S. V. Chmutov

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Abstract: I would like to present a joint work with Clark Butler [2] about a relations between some polynomial invariants of graphs on surfaces and planar graphs.
A famous graph invariant, the Tutte polynomial, was generalized to topological setting of graphs on surfaces by B. Bollobás and O. Riordan in [1] and to relative plane graphs by Y. Diao and G. Hetyei in [3]. We found a relation between these polynomials for graphs obtained by the construction below.
Graphs on surfaces can be studied in terms of plane graphs via their projections preserving the rotation systems. For non-planar graphs such a projection will have singularities. The simplest singularities are double points on edges of the graph. Using them we supplement the image of the graph with some additional edges and vertices. Thus we obtain a relative plane graph which is a plane graph with a distinguished subset of edges.
This relation has an application in knot theory. The classical Thistlethwaite theorem relates the Jones polynomial of a link to the Tutte polynomial of a plane graph obtained from a checkerboard coloring of the regions of the link diagram. Our relation conforms two generalizations of the Thistlethwaite theorem to virtual links from [3,4].

Language: English

  1. B. Bollobás, O. Riordan, “A polynomial of graphs on surfaces”, Math. Ann, 323 (2002), 81 – 96  crossref  mathscinet  zmath  isi  scopus
  2. C. Butler, S. Chmutov, “Bollobás-Riordan and relative Tutte polynomials”, arXiv: 1011.0072
  3. Y. Diao, G. Hetyei, “Relative Tutte polynomials for colored graphs and virtual knot theory”, Combinatorics Probability and Computing, 19 (2010), 343–369  crossref  mathscinet  zmath  isi  scopus
  4. S. Chmutov, “Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial”, Journal of Combinatorial Theory, Ser. B, 99:3 (2009), 617–638  crossref  mathscinet  zmath  isi  scopus

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