Petrie duality and the Anstee-Robertson graph

We define the operation of Petrie duality for maps, describing its general properties both geometrically and algebraically. We give a number of examples and applications, including the construction of a pair of regular maps, one orientable of genus 17, the other non-orientable of genus 52, which embed the 40-vertex cage of valency 6 and girth 5 discovered independently by Robertson and Anstee. We prove that this map (discovered by Evans) and its Petrie dual are the only regular embeddings of this graph, together with a similar result for a graph of order 40, valency 6 and girth 3 with the same automorphism group.