Embeddings of Four-valent Framed Graphs into 2-surfaces

  • Vassily Olegovich ManturovEmail author
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 1)


It is well known that the problem of detecting the least (highest) genus of a surface where a given graph can be embedded is closely connected to the problem of embedding special four-valent framed graphs, i.e. 4-valent graphs with opposite edge structure at vertices specified. This problem has been studied, and some cases (e.g., recognizing planarity) are known to have a polynomial solution.

The aim of the present survey is to connect the problem above to several problems which arise in knot theory and combinatorics: Vassiliev invariants and weight systems coming from Lie algebras, Boolean matrices etc., and to give both partial solutions to the problem above and new formulations of it in the language of knot theory.


Intersection Graph Weight System Virtual Link Klein Bottle Reidemeister Move 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BN95]
    Bar-Natan, D.: On the Vassiliev knot invariants. Topology 34, 423–472 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Bou94]
    Bouchet, A.: Circle graph obstructions. J. Comb. Theory B 60, 107–144 (1994) zbMATHMathSciNetGoogle Scholar
  3. [CSM04]
    Campoamor-Stursberg, R., Manturov, V.O.: Invariant tensors formulae via chord diagrams. J. Math. Sci. 128(4), 3018–3029 (2004) CrossRefMathSciNetGoogle Scholar
  4. [CL07]
    Chmutov, S.V., Lando, S.K.: Mutant knots and intersection graphs. 0704.1313 [math.GT] (2007)
  5. [CDL94]
    Chmutov, S.V., Duzhin,  S.V., Lando, S.K.: Vassiliev knot invariants I–III. Adv. Sov. Math. 21, 117–147 (1994) MathSciNetGoogle Scholar
  6. [CDM]
    Chmutov, S.V., Duzhin, S., Mostovoy, Y.: A draft version of the book about Vassiliev knot invariants. Google Scholar
  7. [CR01]
    Crapo, H., Rosenstiehl, P.: On lacets and their manifolds. Discrete Math. 233, 299–320 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  8. [DFK+06]
    Dasbach, O., Futer, D., Kalfagianni, E., Lin, X.-S., Stoltzfus, N.: The Jones polynomial and graphs on surfaces. math.GT/0605571v3 (2006)
  9. [FMR79]
    Filotti, I.S., Miller, G.L., Reif, J.H.: On determining the genus of a graph in O(vo(g)) steps. In: Proceedings of the 11th Annual Symposium on Theory of Computing, pp. 27–37. ACM, New York (1979) Google Scholar
  10. [Fom91]
    Fomenko, A.T.: The theory of multidimensional integrable hamiltonian systems (with arbitrary many degrees of freedom). Molecular table of all integrable systems with two degrees of freedom. Adv. Sov. Math., (6), 1–35, 1991 Google Scholar
  11. [GPV00]
    Goussarov, M., Polyak, M., Viro, O.: Finite type invariants of classical and virtual knots. Topology 39, 1045–1068 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  12. [IM09a]
    Ilyutko, D.P., Manturov, V.O.: Graph-links. Dokl. Math. 80(2), 739–742 (2009) CrossRefGoogle Scholar
  13. [IM09b]
    Ilyutko, D.P., Manturov, V.O.: Introduction to graph-link theory. J. Knot Theory Ramif. 18(6), 791–823 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  14. [Jon85]
    Jones, V.F.R.: A polynomial invariant for links via Neumann algebras. Bull. Am. Math. Soc. 129, 103–112 (1985) CrossRefGoogle Scholar
  15. [Kau87]
    Kauffman, L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Kau99]
    Kauffman, L.H.: Virtual knot theory. Eur. J. Comb. 20(7), 662–690 (1999) CrossRefMathSciNetGoogle Scholar
  17. [KR08]
    Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. Fundam. Math. 199(1), 191 (2008). math.GT/0401268 CrossRefMathSciNetGoogle Scholar
  18. [KR]
    Khovanov, M., Rozansky, L.: Matrix factorizations and link homology II. math.GT/0505056
  19. [Lan06]
    Lando, S.K.: J-invariants of ornaments and framed chord diagrams. Funct. Anal. Appl. 40(1), 1–13 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  20. [LZ03]
    Lando, S.K., Zvonkin, A.K.: Embedded Graphs. Springer, Berlin (2003) Google Scholar
  21. [LRS87]
    Lins, S., Richter, B., Schank, H.: The gauss code problem off the plane. Aequ. Math., 81–95 (1987) Google Scholar
  22. [LM76]
    Lovász, L., Marx, M.: A forbidden substructure characterization of gauss codes. Acta Sci. Math. (Szeged) 38(12), 115–119 (1976) zbMATHMathSciNetGoogle Scholar
  23. [Low07]
    Lowrance, A.: On knot Floer width and Turaev genus. 0709.0720v1 [math.GT]
  24. [Man00]
    Manturov, V.O.: Bifurcations, atoms, and knots. Mosc. Univ. Math. Bull. 1, 3–8 (2000) MathSciNetGoogle Scholar
  25. [Man02]
    Manturov, V.O.: Chord diagrams and invariant tensors. In: Proceedings of the 1 Colloquium on Lie Theory and Applications, pp. 117–125 (2002) Google Scholar
  26. [Man05b]
    Manturov, V.O.: The proof of Vassiliev’s conjecture on planarity of singular graphs. Izv. Math. 69(5), 169–178 (2005) CrossRefMathSciNetGoogle Scholar
  27. [Man]
    Manturov, V.O.: Minimal diagrams of classical knots. math.GT/0501510
  28. [MU05a]
    Manturov, V.O., Uzlov, T.: Knot Theory. RCD, Izhevsk (2005). (In Russian) Google Scholar
  29. [Mel00]
    Mellor, B.: Three weight systems arising from intersection graphs. math.GT/0004080v1 (2000)
  30. [MM65]
    Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. Math., 211–264 (1965) Google Scholar
  31. [Ros99]
    Rosenstiehl, P.: A new proof of the Gauss interlace conjecture. Adv. Appl. Math. 23, 3–13 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  32. [RR78]
    Rosenstiehl, P., Reed, R.: On the principal edge tripartition of a graph. Ann. Discrete Math. 3, 195–226 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  33. [Sob01]
    Soboleva, E.: Vassiliev knot invariants coming from Lie algebras and 4-invariants. J. Knot Theory Ramif. 10(1), 161–169 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  34. [Tur87]
    Turaev, V.G.: A simple proof of the Murasugi and Kauffman theorems on alternating links. Enseign. Math. 33, 203–225 (1987) zbMATHMathSciNetGoogle Scholar
  35. [Vas90]
    Vassiliev, V.A.: Cohomology of knot spaces. In: Theory of Singularities and Its Applications. Advances in Soviet Mathematics, vol. 1, pp. 23–70. AMS, Providence (1990) Google Scholar
  36. [Vas05]
    Vassiliev, V.A.: First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in ℝn. Izv. Math. 69(5), 865–912 (2005) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.People’s Friendship University of RussiaMoscowRussia

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