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Twisted duality for embedded graphs

Authors: Joanna A. Ellis-Monaghan and Iain Moffatt
Journal: Trans. Amer. Math. Soc. 364 (2012), 1529-1569
MSC (2010): Primary 05C10; Secondary 05C31
Published electronically: October 24, 2011
MathSciNet review: 2869185
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Abstract: We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge, and taking the partial dual with respect to the edge. These two operations give rise to an action of $ {S_3}^{\vert E(G)\vert}$, the ribbon group, on $ G$. The action of the ribbon group on embedded graphs extends the concepts of duality, partial duality, and Petrie duality. We show that this ribbon group action gives a complete characterization of duality in that if $ G$ is any cellularly embedded graph with medial graph $ G_m$, then the orbit of $ G$ under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to $ G_m$. We provide characterizations of special sets of twisted duals (such as the partial duals) of embedded graphs in terms of medial graphs, and we show how different kinds of graph isomorphism give rise to these various notions of duality. The ribbon group action then leads to a deeper understanding of the properties of, and relationships among, various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action.

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Additional Information

Joanna A. Ellis-Monaghan
Affiliation: Department of Mathematics, Saint Michael’s College, 1 Winooski Park, Colchester, Vermont 05439

Iain Moffatt
Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688

Received by editor(s): April 21, 2010
Received by editor(s) in revised form: October 12, 2010, and December 17, 2010
Published electronically: October 24, 2011
Additional Notes: The work of the first author was supported by the National Science Foundation (NSF) under grant number DMS-1001408, by the Vermont Space Grant Consortium through the National Aeronautics and Space Administration (NASA), and by the Vermont Genetics Network through Grant Number P20 RR16462 from the INBRE Program of the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH). This paper’s contents are solely the responsibility of the authors and do not necessarily represent the official views of the NSF, NASA, NCRR, or NIH
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.