The Slit-Slide-Sew

Counting maps on surfaces — graphs embedded in topological spaces — is a classical problem in combinatorics. Bipartite maps (where faces alternate between two types) are often easier to count than general maps. The question is whether there’s a direct structural correspondence between the two families.

Bettinelli and Korkotashvili construct one. Their slit-slide-sew bijection works by cutting along a noncontractible loop on the surface, shifting one position, and reattaching. The operation transforms the map’s structure along the cut — specifically, it changes the parity of lengths for other loops that cross the cut. Applied carefully, this converts a general map into a bipartite one and back.

The through-claim: the relationship between general and bipartite maps is a rotation along a topological loop. Not a global transformation, not a recursive decomposition, but a local geometric operation — slit, slide one step, sew — that changes precisely the parity structure that distinguishes the two families.

For unicellular toroidal maps (single-face maps on a torus), the noncontractible loop structure is simple enough that the bijection gives a complete correspondence. Every general map pairs with a bipartite map through this single operation. The torus has just the right amount of topological complexity — enough noncontractible loops to make the problem interesting, few enough that the bijection is clean.

The relationship between generality and bipartiteness is one stitch of displacement along a loop.