Annals of Combinatorics

, Volume 15, Issue 4, pp 675–706 | Cite as

Expansions for the Bollobás-Riordan Polynomial of Separable Ribbon Graphs

Article

Abstract

We define 2-decompositions of ribbon graphs, which generalize 2-sums and tensor products of graphs. We give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.

Mathematics Subject Classification

05C31 05C10 57M15 

Keywords

ribbon graph embedded graph 2-sum tensor product Tutte polynomial Bollobás-Riordan polynomial Ribbon graph polynomial Jones polynomial 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bollobás B.: Modern Graph Theory. Springer-Verlag, New York (1998)CrossRefMATHGoogle Scholar
  2. 2.
    Bollobás B., Riordan O.: A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc 83, 513–531 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bollobás B., Riordan O.: A polynomial of graphs on surfaces. Math. Ann. 323(1), 81–96 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brylawski T.H.: The Tutte polynomial I: general theory. In: Barlotti, A. (eds) Matroid Theory and Its Applications., pp. 125–275. Liguori, Naples (1982)Google Scholar
  5. 5.
    Brylawski T.H., Oxley J.G.: The Tutte polynomial and its applications. In: White, N. (eds) Matroid Applications., pp. 123–225. Cambridge Univ. Press, Cambridge (1992)CrossRefGoogle Scholar
  6. 6.
    Chmutov S., Pak I.: The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial. Mosc. Math. J. 7(3), 409–418 (2007)MathSciNetMATHGoogle Scholar
  7. 7.
    Chmutov S., Voltz J.: Thistlethwaite’s theorem for virtual links. J. Knot Theory Ramifications 17(10), 1189–1198 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dasbach O.T., Futer D., Kalfagianni E., Lin X.-S., Stoltzfus N.W.: The Jones polynomial and graphs on surfaces. J. Combin. Theory Ser. B 98(2), 384–399 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Goldberg L.A., Jerrum M.: Inapproximability of the Tutte polynomial. Inform. and Comput. 206(7), 908–929 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gross J.L., Tucker T.W.: Topological Graph Theory. John Wiley & Sons, Inc., New York (1987)MATHGoogle Scholar
  11. 11.
    Huggett S.: On tangles and matroids. J. Knot Theory Ramifications 14(7), 919–929 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jaeger F.: Tutte polynomials and link polynomials. Proc. Amer. Math. Soc 103(2), 647–654 (1988)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jaeger F., Vertigan D.L., Welsh D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc 108(1), 35–53 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jerrum, M.: Approximating the Tutte polynomial. In: Grimmett, G.,McDiarmid, C. (eds.) Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh, pp. 144–161. Oxford University Press, Oxford (2007)Google Scholar
  15. 15.
    Kauffman L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Loebl M., Moffatt I.: The chromatic polynomial of fatgraphs and its categorification. Adv. Math 217(4), 1558–1587 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Moffatt I.: Knot invariants and the Bollobás-Riordan polynomial of embedded graphs. European J. Combin 29(1), 95–107 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Seymour P.D.: Decomposition of regular matroids. J. Combin. Theory Ser. B 28(3), 305–359 (1980)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Webb, B.S. (ed.) Surveys in Combinatorics, 2005, pp. 173–226. Cambridge Univ. Press, Cambridge (2005)Google Scholar
  20. 20.
    Thistlethwaite M.B.: A spanning tree expansion of the Jones polynomial. Topology 26(3), 297–309 (1987)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Traldi L.: A dichromatic polynomial for weighted graphs and link polynomials. Proc. Amer. Math. Soc 106(1), 279–286 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Woodall D.R.: Tutte polynomial expansions for 2-separable graphs. Discrete Math. 247(1-3), 201–213 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of PlymouthPlymouthUK
  2. 2.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

Personalised recommendations