Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial
نویسندگان
چکیده
منابع مشابه
Quasi-tree expansion for the Bollobás--Riordan--Tutte polynomial
Bollobás and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are discs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing Tutte’s concept of activity to quasi-tree...
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We extend the quasi-tree expansion of A. Champanerkar, I. Kofman, and N. Stoltzfus to not necessarily orientable ribbon graphs. We study the duality properties of the Bollobás-Riordan polynomial in terms of this expansion. As a corollary, we get a “connected state” expansion of the Kauffman bracket of virtual link diagrams. Our proofs use extensively the partial duality of S. Chmutov.
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Ribbon graphs are surfaces with boundary together with a decomposition into a union of closed topological discs of two types, edges and vertices. These sets are subject to some natural axioms recalled in section 2.1. For such a generalisation of the usual graphs, B. Bollobás and O. Riordan found a topological version of the Tutte polynomial [3, 4]. In the following, we will refer to this genera...
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In [BR01], [BR02], Bollobás and Riordan generalized the classical Tutte polynomial to graphs cellularly embedded in surfaces, i.e. ribbon graphs, thus encoding topological information not captured by the classical Tutte polynomial. We provide a ‘recipe theorem’ for their new topological Tutte polynomial, R(G). We then relate R(G) to the generalized transition polynomial Q(G) of [E-MS02] via a m...
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The coloured Tutte polynomial by Bollobás and Riordan is, as a generalisation of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines. We establish a similar...
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ژورنال
عنوان ژورنال: Bulletin of the London Mathematical Society
سال: 2011
ISSN: 0024-6093
DOI: 10.1112/blms/bdr034