Abstract
In Chap. 3, we address two fundamental questions: (1) What is the complete twisted duality analog to the relation between Tait graphs and geometric duality? (2) How is a hierarchy of graph equivalences captured by a hierarchy of twisted dualities? We construct cycle family graphs and show that they fully characterise all twisted duals with a given (abstract) medial graph, and use this to answer Question 1. For Question 2, we give a hierarchy of graph structures, ranging from abstract graphs, through cyclically ordered graphs, to embedded graphs. We show that when we consider medial graphs up to equivalence under these varying levels of specificity, then there is a corresponding hierarchy of duality, ranging from twisted duality, through partial duality, to geometric duality.
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