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تسجيل مستخدم جديدEvaluations of topological Tutte polynomials
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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 Bollobas and Riordans 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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We introduce and study the notion of the $G$-Tutte polynomial for a list $mathcal{A}$ of elements in a finitely generated abelian group $Gamma$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite
abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Branden-Mocis arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex. As in the classical case, $G$-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting and the Poincare polynomial) of abelian Lie group arrangements. We also discuss differences between the arithmetic Tutte and the $G$-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients.
In [A polynomial invariant of graphs on orientable surfaces, Proc. Lond. Math. Soc., III Ser. 83, No. 3, 513-531 (2001)] and [A polynomial of graphs on surfaces, Math. Ann. 323, 81-96 (2002)], Bollobas and Riordan generalized the classical Tutte poly
nomial 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) via a medial graph construction, thus extending the relation between the classical Tutte polynomial and the Martin, or circuit partition, polynomial to ribbon graphs. We use this relation to prove a duality property for R(G) that holds for both oriented and unoriented ribbon graphs. We conclude by placing the results of Chumutov and Pak [The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs, Moscow Mathematical Journal 7(3) (2007) 409-418] for virtual links in the context of the relation between R(G) and Q(R).
A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobas-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-va
luations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.
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