Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-30T19:30:44.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Graphs, Links, and Duality on Surfaces

Published online by Cambridge University Press:  29 September 2010

VYACHESLAV KRUSHKAL
VYACHESLAV KRUSHKAL*
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a polynomial invariant of graphs on surfaces, PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for PG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, PG specializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomial PG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 267 - 287
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bollobás, B. (1998) Modern Graph Theory, Springer.CrossRefGoogle Scholar
[2]Bollobás, B. and Riordan, O. (2001) A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc. (3) 83 513–531.CrossRefGoogle Scholar
[3]Bollobás, B. and Riordan, O. (2002) A polynomial of graphs on surfaces. Math. Ann. 323 81–96.Google Scholar
[4]Chmutov, S. (2009) Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial. J. Combin. Theory Ser. B 99 617–638.CrossRefGoogle Scholar
[5]Chmutov, S. and Pak, I. (2007) The Kauffman bracket of virtual links and the Bollobás–Riordan polynomial. Mosc. Math. J. 7 409–418.CrossRefGoogle Scholar
[6]Costa-Santos, R. and McCoy, B. M. (2002) Dimers and the critical Ising model on lattices of genus > 1. Nuclear Phys. B 623 439–473.CrossRefGoogle Scholar
[7]Dasbach, O. T., Futer, D., Kalfagianni, E., Lin, X.-S. and Stoltzfus, N. W. (2008) The Jones polynomial and graphs on surfaces. J. Combin. Theory Ser. B 98 384–399.Google Scholar
[8]Di Francesco, P., Saleur, H. and Zuber, J.-B. (1987) Critical Ising correlation functions in the plane and on the torus. Nuclear Phys. B 290 527–581.CrossRefGoogle Scholar
[9]Dye, H. and Kauffman, L. H. (2005) Minimal surface representations of virtual knots and links. Algebr. Geom. Topol. 5 509–535.CrossRefGoogle Scholar
[10]Ellis-Monaghan, J. A. and Sarmiento, I. (2005) A duality relation for the topological Tutte polynomial. Talk at the AMS Eastern Section Meeting #1009, on Graph and Matroid Invariants, Bard College. http://academics.smcvt.edu/jellis-monaghanGoogle Scholar
[11]Ellis-Monaghan, J. A. and Sarmiento, I. A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan. European J. Combin., to appear.Google Scholar
[12]Fendley, P. and Krushkal, V. (2009) Tutte chromatic identities from the Temperley–Lieb algebra. Geometry & Topology 13 709–741.CrossRefGoogle Scholar
[13]Fendley, P. and Krushkal, V. (2010) Link invariants, the chromatic polynomial and the Potts model. Adv. Theor. Math. Phys. 14 507–540.CrossRefGoogle Scholar
[14]Hatcher, A. (2002) Algebraic Topology, Cambridge University Press.Google Scholar
[15]Jacobsen, J. L. and Saleur, H. (2008) Conformal boundary loop models. Nucl. Phys. B 788 137–166.CrossRefGoogle Scholar
[16]Jaeger, F. (1988) Tutte polynomials and link polynomials. Proc. Amer. Math. Soc. 103 647–654.CrossRefGoogle Scholar
[17]Jones, V. F. R. (1985) A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 103–111.CrossRefGoogle Scholar
[18]Kamada, N. (2002) On the Jones polynomials of checkerboard colorable virtual knots. Osaka J. Math. 39 325–333.Google Scholar
[19]Kauffman, L. H. (1987) State models and the Jones polynomial. Topology 26 395–407.Google Scholar
[20]Kauffman, L. H. (1999) Virtual knot theory. Europ. J. Combin. 20 663–690.CrossRefGoogle Scholar
[21]Kuperberg, G. (2003) What is a virtual link? Algebr. Geom. Topol. 3 587–591.CrossRefGoogle Scholar
[22]Manturov, V. O. (2003) Kauffman-like polynomial and curves in 2-surfaces. J. Knot Theory Ramifications 12 1131–1144.CrossRefGoogle Scholar
[23]Massey, W. S. (1991) A Basic Course in Algebraic Topology, Vol. 127 of Graduate Texts in Mathematics, Springer.Google Scholar
[24]Moffatt, I. (2008) Knot invariants and the Bollobás–Riordan polynomial of embedded graphs. Europ. J. Combin. 29 95–107.CrossRefGoogle Scholar
[25]Moffatt, I. (2010) Partial duality and Bollobás and Riordan's ribbon graph polynomial. Discrete Math. 310 174–183.CrossRefGoogle Scholar
[26]Morin-Duchesne, A. and Saint-Aubin, Y. (2009) Critical exponents for the homology of Fortuin–Kasteleyn clusters on a torus. Phys. Rev. E 80, 021130.Google Scholar
[27]Przytycki, J. (1991) Skein modules of 3-manifolds. Bull. Polish Acad. Sci. Math. 39 91–100.Google Scholar
[28]Sokal, A. D. (2005) The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In Surveys in Combinatorics 2005 (Webb, B. S., ed.), Cambridge University Press, pp. 173–226.CrossRefGoogle Scholar
[29]Thistlethwaite, M. B. (1987) A spanning tree expansion for the Jones polynomial. Topology 26 297–309.Google Scholar
[30]Turaev, V. (1988) The Conway and Kauffman modules of a solid torus. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 79–89. Translation in J. Soviet Math. 52 (1990), 2799–2805.Google Scholar
[31]Tutte, W. T. (1947) A ring in graph theory. Proc. Cambridge Phil. Soc. 43 26–40.CrossRefGoogle Scholar
[32]Tutte, W. T. (1954) A contribution to the theory of chromatic polynomials. Canad. J. Math. 6 80–91.Google Scholar
[33]Vignes-Tourneret, F. (2009) The multivariate signed Bollobas–Riordan polynomial. Discrete Math. 309 5968–5981.Google Scholar