Bollobás–Riordan and Relative Tutte Polynomials

Butler, Clark; Chmutov, Sergei

doi:10.1007/s40598-015-0021-7

Arnold Math J.

2015

DOI: 10.1007/s40598-015-0021-7

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Bollobás–Riordan and Relative Tutte Polynomials

Clark Butler

Sergei Chmutov

Abstract: We establish a relation between the Bollobás-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.

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“…As expected by Section 4.1, the invariant T SF (M ) is "essentially" bivariate, with a bivariate prefactor. Also, by applying the inverse of the above morphism U (j) and recollecting like variables, we could make the exponents in formula (14) identical to those in the universal Tutte character for matroids, the corank-nullity generating function (3). However, this rearrangement is not especially motivated for submodular functions and polymatroids, since |A| − rk(A) is not guaranteed to be nonnegative.…”

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Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

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This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.obtain several convolution formulae for different classes of combinatorial objects. Some of these formulae are well known, while others are (to the best of our knowledge) new: see for instance Propositions 5.12, 5.13 and 5.14 for the classical Tutte polynomial, Proposition 6.17 for the Las Vergnas polynomial, Proposition 6.20 for the Bollobás-Riordan polynomial and Theorem 10.9 for the arithmetic Tutte polynomial.We have chosen to exemplify our results with mostly matroid-like combinatorial structures, which are arguably simpler to deal with than topological examples. Our formalism, and in particular the mechanism of twist maps, should help uncover new invariants in the latter class and produce interesting convolution formulae. This will be the subject of a subsequent article.Layout. The structure of this paper is as follows. In Section 2 we give the definitions of minors systems and comonoids in set species. Section 3 contains the main results, including the definitions and statements of our universal invariants and formulae. Section 4 presents a number of further results which are less essential for our main developments.The remainder of the paper, Sections 5 through 10, comprises a sequence of applications of our theory to numerous individual minors systems. We work out Grothendieck groups, universal Tutte characters, and in many cases universal convolution formulae, and point out how these specialize to invariants and formulae present in the literature. Section 5 covers the minors system of matroids, of which all the subsequent sections are in one way or another generalizations, together with the minors system of graphs. Of the following sections, Section 6 is on delta-matroids, matroid perspectives, and their ilk is called on in Section 7 on relative matroids, but there are no (or at most incidental) dependences between these and Section 8 on polymatroids and generalised permutohedra, Section 9 on colored matroids, or Section 10 on arithmetic matroids, nor among the latter three, so the reader should have no trouble taking these in any order. The last four sections are also new by comparison with [37].Notation. We fix a commutative ring with unit K...

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Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

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Bollobás–Riordan and Relative Tutte Polynomials

Cited by 1 publication

References 29 publications

Universal Tutte characters via combinatorial coalgebras

Universal Tutte characters via combinatorial coalgebras

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