Abstract
This chapter introduces the framework upon which we will build the rest of this monograph. We begin by considering geometric duals and Petrials, observing that Petriality and geometric duality result from local operations on each edge of an embedded graph. These local operations applied to subsets of the edge set result in partial Petrality and partial duality. We provide constructions for partial duals and partial Petrials in various realisations of embedded graphs.The two operations of partial Petrality and partial duality give rise to an action of the symmetric group on embedded graphs with a distinguished set of edges. This group action leads to twisted duality, which assimilates several types of duality from the literature, including geometric duality, direct derivatives, Petrie duals, and partial duality. We concluded by defining the ribbon group and describing how twisted duality can be obtained as orbits under the ribbon group action on the set of edge-ordered embedded graphs.
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