Twisted duality for embedded graphs

Ellis-Monaghan, Joanna A.; Moffatt, Iain

doi:10.1090/s0002-9947-2011-05529-7

Cited by 45 publications

(100 citation statements)

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“…Here we define partial duals directly on ribbon graphs. We refer the reader to [22,31,50] or the exposition [32] for alternative constructions and other perspectives of partial duals.…”

Section: Geometric Duals and Partial Dualsmentioning

confidence: 99%

Matroids, delta-matroids and embedded graphs

Chun

Moffatt

Noble

et al. 2019

Journal of Combinatorial Theory, Series A

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Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illustrate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the literature, including the Las Vergnas, Bollabás-Riordan and Krushkal polynomials, are in fact delta-matroidal. (Carolyn Chun), iain.moffatt@rhul.ac.uk (Iain Moffatt), steven.noble@brunel.ac.uk (Steven D. Noble), ralf@rueckriemen.de (Ralf Rueckriemen)1 Ralf Rueckriemen was financed by the DFG through grant RU 1731/1-1.

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“…Here we define partial duals directly on ribbon graphs. We refer the reader to [22,31,50] or the exposition [32] for alternative constructions and other perspectives of partial duals.…”

Section: Geometric Duals and Partial Dualsmentioning

confidence: 99%

Matroids, delta-matroids and embedded graphs

Chun

Moffatt

Noble

et al. 2019

Journal of Combinatorial Theory, Series A

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“…The well-known universality property of the Tutte polynomial of a matroid can be formulated as saying that there exists a unique, well-defined, matroid polynomial f M (x, y, a, b) given by (19) f…”

Section: Universal Formsmentioning

confidence: 99%

“…The final problem is about the Bollobás-Riordan polynomial R G (x, y, z). Most of the known results about this polynomial, particularly its combinatorial interpretations, do not apply to the full 3-variable polynomial R G (x, y, z), but rather to its 2-variable specialisation x γ(G)/2 R G (x + 1, y, 1/ √ xy) (see, for example, [7,12,16,19,23,24,29] 4.9. The Penrose polynomial as a Tutte polynomial.…”

Section: Theorem 9 Givesmentioning

confidence: 99%

Hopf algebras and Tutte polynomials

Krajewski

Moffatt

Tanasă

2018

Advances in Applied Mathematics

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By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobás and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial.We show that our Tutte polynomials of Hopf algebras share common properties with the classical Tutte polynomial, including deletion-contraction definitions, universality properties, convolution formulas, and duality relations. New results for graph polynomials from the literature are then obtained as examples of the general results.Our results offer a framework for the study of the Tutte polynomial and its analogues in other settings, offering the means to determine the properties and connections between a wide class of polynomial invariants.

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“…It was introduced to unify various versions of Thistlethwaite theorems [2][3][4] in knot theory that relate the Jones polynomial of knots with a Tutte-like polynomial of graphs. Partial duality was further generalized to twisted duality by Ellis-Monaghan and Moffatt in [5]. Both are far-reaching extensions of geometric duality and have found a number of significant applications in graph theory, topology, and physics (see, e.g., [7][8][9][10][11][12][13][14]).…”

Section: Introductionmentioning

confidence: 99%