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Unsigned State Models for the Jones Polynomial
Iain Moffatt
Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West Waterloo, Ontario, Canada
imoffatt@jaguar1.usouthal.edu
Annals of Combinatorics 15 (1) pp.127-146 January, 2011
AMS Subject Classification: 05C10, 57M27
Abstract:
It is a well-known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in S3 there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating to the Jones and Bollob´as-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.
Keywords: Bollobás-Riordan polynomial, embedded graphs, Jones polynomial, medial graph, Potts model, ribbon graphs, vertex model

References:

1. Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press, Inc., London (1982)

2. Bollob´as, B., Riordan, O.: A polynomial for graphs on orientable surfaces. Proc. London Math. Soc. (3) 83, 513--531 (2001)

3. Bollob´as, B., Riordan, O.: A polynomial of graphs on surfaces. Math. Ann. 323(1), 81--96 (2002)

4. Chmutov, S., Pak, I.: The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs. arXiv:math.GT/0404475 (2004)

5. Chmutov, S., Pak, I.: The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial. Mosc. Math. J. 7, 409--418 (2007)

6. Chmutov, S., Voltz, J.: Thistlethwaite's theorem for virtual links. J. Knot Theory Ramf cations 17(10), 1189--1198 (2008)

7. 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)

8. Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model, I. Introduction and relation to other models. Physica 57, 536--564 (1972)

9. Huggett, S., Moffatt, I., Expansions for the Bollobás-Riordan polynomial of separable ribbon graphs. Ann. Combin. (to appear)

10. Inoue, K., Kaneto, T.: A Jones type invariant of links in the product space of a surface and the real line. J. Knot Theory Ramifications 3(2), 153--161 (1994)

11. Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12(1), 103--111 (1985)

12. Jones, V.F.R.: On knot invariants related to some statistical mechanical models. Pacific J. Math. 137(2), 311--334 (1989)

13. Kauffman, L.H., State models and the Jones polynomial. Topology 26(3), 395--407 (1987)

14. Kauffman, L.H., Statistical mechanics and the Jones polynomial. Amer. Math. Soc., Providence, RI (1988)

15. Kauffman, L.H.: A Tutte polynomial for signed graphs. Discrete Appl. Math. 25(1-2), 105--127 (1989)

16. Loebl, M., Moffatt, I.: The chromatic polynomial of fatgraphs and its categorification. Adv. Math. 217(4), 1558--1587 (2008)

17. Loebl, M., Chromatic polynomial, q-binomial counting and colored Jones function. Adv. Math. 211(2), 546--565 (2007)

18. Moffatt, I.: Knot invariants and the Bollob´as-Riordan polynomial of embedded graphs. European J. Combin. 29(1), 95--107 (2008)

19. Perk, J.H.H., Wu, F.Y.: Graphical approach to the nonintersecting string model: startriangle equation, inversion relation, and exact solution. Phys. A 138(1-2), 100--124 (1986)

20. Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., Vol. 327, pp. 173--226. Cambridge Univ. Press, Cambridge (2005)

21. Thistlethwaite, M.B.: A spanning tree expansion of the Jones polynomial. Topology 26(3), 297--309 (1987)

22. Turaev, V.G.: The Yang-Baxter equation and invariants of links. Invent. Math. 92(3), 527-- 553 (1988)

23. Wu, F.Y.: Knot theory and statistical mechanics. Rev. Modern Phys. 64(4), 1099--1131 (1992)

24. Wu, F.Y.: Jones polynomial as a Potts model partition function. J. Knot Theory Ramifications 1(1), 47--57 (1992)