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Interactions with Graph Polynomials

  • Joanna A. Ellis-Monaghan
  • Iain Moffatt
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Chapter 4 explores twisted duality as a tool for extracting both combinatorial and topological information from topological graph polynomials. We begin with the topological transition polynomial of Ellis-Monaghan and Moffatt (Trans. Amer. Math. Soc., 364, 1529–1569, 2012), which interacts with twisted duality in a particularly natural way, leading to a generalised duality identity, and a three term contraction-deletion relation. The topological transition polynomial specialises to the topological Penrose polynomial and, with some restriction, agrees with the topochromatic polynomial and hence the ribbon graph polynomial of Bollobás and Riordan. Thus, the identities for the topological transition polynomial lead in turn to new results for these polynomials, including reformulations of the Four Colour Theorem. We survey these results in this chapter.

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Copyright information

© Joanna A. Ellis-Monaghan, Iain Moffatt 2013

Authors and Affiliations

  • Joanna A. Ellis-Monaghan
    • 1
  • Iain Moffatt
    • 2
  1. 1.Department of MathematicsSaint Michael’s CollegeColchesterUSA
  2. 2.Department of MathematicsRoyal Holloway University of LondonEgham, SurreyUK

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