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Evaluations of Topological Tutte Polynomials

  • J. ELLIS-MONAGHAN (a1) and I. MOFFATT (a2)

We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

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[1] M. Aigner (1997) The Penrose polynomial of a plane graph. Math. Ann. 307 173189.

[3] B. Bollobás and O. Riordan (2001) A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc. 83 513531.

[4] B. Bollobás and O. Riordan (2002) A polynomial of graphs on surfaces. Math. Ann. 323 8196.

[6] S. Chmutov (2009) Generalized duality for graphs on surfaces and the signed Bollobás–Riordan poly-nomial. J. Combin. Theory Ser. B 99 617638.

[8] O. T. Dasbach , D. Futer , E. Kalfagianni , X.-S. Lin and N. W. Stoltzfus (2008) The Jones polynomial and graphs on surfaces. J. Combin. Theory Ser. B 98 384399.

[9] J. A. Ellis-Monaghan (1998) New results for the Martin polynomial. J. Combin. Theory Ser. B 74 326352.

[10] J. A. Ellis-Monaghan and I. Moffatt (2012) Twisted duality for embedded graphs. Trans. Amer. Math. Soc. 364 15291569.

[11] J. A. Ellis-Monaghan and I. Moffatt (2013) A Penrose polynomial for embedded graphs. European J. Combin. 34 424445.

[13] J. A. Ellis-Monaghan and I. Sarmiento (2011) A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan. European J. Combin. 32 782794.

[14] S. Huggett and I. Moffatt (2011) Expansions for the Bollobás–Riordan and Tutte polynomials of separable ribbon graphs. Ann. Comb. 15 675706.

[15] F. Jaeger (1988) On Tutte polynomials and cycles of plane graphs. J. Combin. Theory Ser. B 44 127146.

[16] F. Jaeger (1990) On transition polynomials of 4-regular graphs. In Cycles and Rays, Vol. 301 of NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci, Kluwer Academic, pp. 123150.

[19] I. Moffatt (2008) Knot invariants and the Bollobás–Riordan polynomial of embedded graphs. European J. Combin. 29 95107.

[20] I. Moffatt (2013) Separability and the genus of a partial dual. European J. Combin. 34 355378.

[21] J. G. Oxley and D. J. A. Welsh (1992) Tutte polynomials computable in polynomial time. Discrete Math. 109 185192.

[23] L. Traldi (2005) Parallel connections and coloured Tutte polynomials. Discrete Math. 290 291299.

[24] S. E. Wilson (1979) Operators over regular maps. Pacific J. Math. 81 559568.

[25] D. R. Woodall (2002) Tutte polynomial expansions for 2-separable graphs. Discrete Math. 247 201213.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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