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Topological crossovers are a class of representation-independent operators that are well-defined once a notion of distance over the solution space is defined. In this paper we explore how the topological framework applies to the permutation representation and in particular analyze the consequences of having more than one notion of distance available. Also, we study the interactions among distances and build a rational picture in which pre-existing recombination/crossover operators for permutation fit naturally. Lastly, we also analyze the application of topological crossover to TSP.