# Normal forms for planar connected string diagrams

In the graphical calculus of planar string diagrams, equality is generated by the left and right exchange moves, which swaps the heights of adjacent vertices. We show that for connected diagrams the left- and right-handed exchanges each give strongly normalizing rewrite strategies. We show that these strategies terminate in O(n^3) steps where n is the number of vertices. We also give an algorithm to directly construct the normal form, and hence determine isotopy, in O(n · m) time, where m is the number of edges.

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