New Dualities From Old: generating geometric, Petrie, and Wilson dualities and trialites of ribbon graphs (original) (raw)
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We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutation group and the original ribbon group of Ellis-Monaghan and Moffatt to take (partial) twists and duals, or twuals, of ribbon graphs. A ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twuals. This extends the original ribbon group action, which only used the canonical identification of edges, to the more natural setting of self-twuality up to isomorphism. We then show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twuality properties of these bouquets propagate through their orbits. Thus, we can determine (partial) twualities via these one vertex graphs, for which checking isomorphism reduces simply to checking dihedral group symmetries. Finally, we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges, in contrast with the...
Characterization of regular checkerboard colourable twisted duals of ribbon graphs
Journal of Combinatorial Theory, Series A, 2021
The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. The twisted dual is a further generalization by combining the partial Petrial. Given a ribbon graph G, in this paper, we first characterize regular partial duals of the ribbon graph G by using spanning quasi-tree and its related shorter marking arrow sequence set. Then we characterize checkerboard colourable partial Petrials for any Eulerian ribbon graph by using spanning trees and a related notion of adjoint set. Finally we give a complete characterization of all regular checkerboard colourable twisted duals of a ribbon graph, which solve a problem raised by Ellis-Monaghan and Moffatt [T. Am. Math. Soc., 364(3) (2012), 1529-1569].
Partial duality for ribbon graphs, I: Distributions
European Journal of Combinatorics, 2020
The partial dual G A with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones-Kauffman and Bollobás-Riordan polynomials, and it has developed into a topic of independent interest. This paper studies, for a given G, the enumeration of the partial duals of G by Euler genus, as represented by its generating function, which we call the partial-dual Euler-genus polynomial of G. A recursion is given for subdivision of an edge and is used to derive closed formulas for the partial-dual genus polynomials of four families of ribbon graphs. The log-concavity of these polynomials is studied in some detail. We include a concise, self-contained proof that χ (G A) = χ (A) + χ (E(G) − A) − 2|V (G)|, where χ(G) = |V (G)| − |E(G)| + |F (G)|, and where A represents the ribbon graph obtained from G by deleting all edges not in A. This formula is a variant of a result of Moffatt.
2011
We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ribbon graph from a virtual link whose extended Bollob\'as-Riordan polynomial specializes to the arrow polynomial of the virtual link recently introduced by H.Dye and L.Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.
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European Journal of Combinatorics, 1999
An overview is provided of some of the basic facts concerning rim hook lattices and ribbon tableaux, using a representation of partitions by their edge sequences. An action is defined of the affine Coxeter group of typeà r−1 on the r-rim hook lattice, and thereby on the sets of standard and semistandard r-ribbon tableaux, and this action is related to the concept of chains in r-ribbon tableaux.
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Linear Algebra and its Applications, 2000
Let G be a signed plane graph and G d its signed dual graph. Methods from knot theory are used to show that the signed Laplacian matrices L(G) and L(G d ) are Goeritz congruent. There exists diagonal (0, ±1)-matrices, 1 and 2 , and a unimodular matrix U such that
On terminal forms for topological polynomials for ribbon graphs: The -petal flower
European Journal of Combinatorics, 2014
The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] extends the Tutte polynomial and its contraction/deletion rule for ordinary graphs to ribbon graphs. Given a ribbon graph G, the related polynomial should be computable from the knowledge of the terminal forms of G namely specific induced graphs for which the contraction/deletion procedure becomes more involved. We consider some classes of terminal forms as rosette ribbon graphs with N ≥ 1 petals and solve their associate Bollobas-Riordan polynomial. This work therefore enlarges the list of terminal forms for ribbon graphs for which the Bollobas-Riordan polynomial could be directly deduced.