The topological Tutte polynomials of Bollobas and Riordan: properties and relations to other graph polynomials
Seminar Room 1, Newton Institute
In [BR01], [BR02], Bollob´as and Riordan defined analogs of the Tutte polynomial for graphs embedded in surfaces, thus encoding topological information lost in the classical Tutte polynomial. We pro- vide a recipe theorem for these polynomials and use it to relate them to the generalized transition polynomial, the topological Tutte poly- nomials defined in [Las75], [Las78], [Las79], the parametrized Tutte polynomial of [Zas92] and [BR99], and Bouchets Tutte-Martin poly- nomial of isotropic systems. Various evaluations of these polynomi- als of Bollob´as and Riordan, as well as insight into the topological information they encode. The relationship between the generalized transition polynomial and the topological Tutte polynomial extends a result of [Jae90] from planar graphs to arbitrary graphs by giving a relationship between the transition and the R polynomials. We also visit the Kauffman bracket in light of these relationships and that es- tablished between it and the topolofical Tutte polynomial in [CP].
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