91 research outputs found

    Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

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    We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric

    Dichromatic polynomials and Potts models summed over rooted maps

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    We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the qq-state Potts model randomized over such maps. Like the regular ferromagnetic lattice models, it has a first-order transition when qq is greater than a critical value qcq_c, but qcq_c is much larger - about 72 instead of 4.Comment: 29 pages, three figures changes in App D, introduction and acknowledgement

    Evaluations of topological Tutte polynomials

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    We find new properties of the topological transition polynomial of embedded graphs, Q(G)Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, R(G)R(G), and the topological Penrose polynomial, P(G)P(G). The general framework provided by Q(G)Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G)P(G), R(G)R(G), and the Tutte polynomial, T(G)T(G), as sums of chromatic polynomials of graphs derived from GG; show that these polynomials count kk-valuations of medial graphs; show that R(G)R(G) counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of R(G)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)P(G) and R(G)R(G).Comment: V2: major revision, several new results, and improved expositio

    The multivariate signed Bollobas-Riordan polynomial

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    We generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak [Moscow Math. J. 7 (2007), no. 3, 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [J. Combinatorial Theory, Ser. B, 99 (3): 617-638, 2009] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.Comment: 17 pages, 2 figures. Published version: a section added about the quasi-tree expansion of the multivariate Bollobas-Riordan polynomia

    Knot invariants and the Bollobas-Riordan polynomial of embedded graphs

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    For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove results about graph polynomials and, after discussing questions of equivalence of the polynomials, we go on to use our formulae to prove a duality relation for the Bollobas-Riordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the Bollobas-Riordan polynomials of an embedded graph and its tensor product with a cycle.Comment: v2: minor corrections, to appear in European Journal of Combinatoric

    Unsigned state models for the Jones polynomial

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    It is well a 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 the 3-sphere 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 the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric

    Degree distribution in random planar graphs

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    We prove that for each k≥0k\ge0, the probability that a root vertex in a random planar graph has degree kk tends to a computable constant dkd_k, so that the expected number of vertices of degree kk is asymptotically dknd_k n, and moreover that ∑kdk=1\sum_k d_k =1. The proof uses the tools developed by Gimenez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=∑kdkwkp(w)=\sum_k d_k w^k. From this we can compute the dkd_k to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qkd_k \sim c\cdot k^{-1/2} q^k for large values of kk, where q≈0.67q \approx 0.67 is a constant defined analytically

    The Tensor Track, III

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    We provide an informal up-to-date review of the tensor track approach to quantum gravity. In a long introduction we describe in simple terms the motivations for this approach. Then the many recent advances are summarized, with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion, Osterwalder-Schrader positivity...) which, while important for the tensor track program, are not detailed in the usual quantum gravity literature. We list open questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
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