A. Ellis-Monaghan, Joanna and Moffatt, Iain (2012) A Penrose polynomial for embedded graphs. European Journal of Combinatorics
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We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem.
This is a Accepted version This version's date is: 2012 This item is not peer reviewed
https://repository.royalholloway.ac.uk/items/d5def4b8-ab65-141b-9f11-bb2c1f52baa7/1/
Deposited by Research Information System (atira) on 17-Sep-2012 in Royal Holloway Research Online.Last modified on 17-Sep-2012