Skip to main content Accessibility help

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).



Hide All
[1]Aigner, M. (1997) The Penrose polynomial of a plane graph. Math. Ann. 307 173189.
[2]Aigner, M. (2000) Die Ideen von Penrose zum 4-Farbenproblem. Jahresber. Deutsch. Math.-Verein. 102 4368.
[3]Bollobás, B. and Riordan, O. (2001) A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc. 83 513531.
[4]Bollobás, B. and Riordan, O. (2002) A polynomial of graphs on surfaces. Math. Ann. 323 8196.
[5]Brylawski, T. (1982) The Tutte polynomial I: General theory. In Matroid Theory and its Applications, Liguori, Naples, pp. 125–275.
[6]Chmutov, S. (2009) Generalized duality for graphs on surfaces and the signed Bollobás–Riordan poly-nomial. J. Combin. Theory Ser. B 99 617638.
[7]Chmutov, S. and Pak, I. (2007) The Kauffman bracket of virtual links and the Bollobás–Riordan poly-nomial. Mosc. Math. J. 7 409418.
[8]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 384399.
[9]Ellis-Monaghan, J. A. (1998) New results for the Martin polynomial. J. Combin. Theory Ser. B 74 326352.
[10]Ellis-Monaghan, J. A. and Moffatt, I. (2012) Twisted duality for embedded graphs. Trans. Amer. Math. Soc. 364 15291569.
[11]Ellis-Monaghan, J. A. and Moffatt, I. (2013) A Penrose polynomial for embedded graphs. European J. Combin. 34 424445.
[12]Ellis-Monaghan, J. A. and Sarmiento, I. (2002) Generalized transition polynomials. Congr. Numer. 155 5769.
[13]Ellis-Monaghan, J. A. and Sarmiento, I. (2011) A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan. European J. Combin. 32 782794.
[14]Huggett, S. and Moffatt, I. (2011) Expansions for the Bollobás–Riordan and Tutte polynomials of separable ribbon graphs. Ann. Comb. 15 675706.
[15]Jaeger, F. (1988) On Tutte polynomials and cycles of plane graphs. J. Combin. Theory Ser. B 44 127146.
[16]Jaeger, F. (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.
[17]Korn, M. and Pak, I. (2003) Combinatorial evaluations of the Tutte polynomial. Preprint.
[18]Las Vergnas, M. (1983) Le polynôme de Martin d'un graphe Eulerien. Ann. Discrete Math. 17 397411.
[19]Moffatt, I. (2008) Knot invariants and the Bollobás–Riordan polynomial of embedded graphs. European J. Combin. 29 95107.
[20]Moffatt, I. (2013) Separability and the genus of a partial dual. European J. Combin. 34 355378.
[21]Oxley, J. G. and Welsh, D. J. A. (1992) Tutte polynomials computable in polynomial time. Discrete Math. 109 185192.
[22]Penrose, R. (1971) Applications of negative dimensional tensors. In Combinatorial Mathematics and its Applications (Welsh, D. J. A., ed.), Academic Press, pp. 221244.
[23]Traldi, L. (2005) Parallel connections and coloured Tutte polynomials. Discrete Math. 290 291299.
[24]Wilson, S. E. (1979) Operators over regular maps. Pacific J. Math. 81 559568.
[25]Woodall, D. R. (2002) Tutte polynomial expansions for 2-separable graphs. Discrete Math. 247 201213.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed