Twisted Surfaces

This article proves the laundry embedding theorem. It considers surface triples (S,G,J) in S ³ where S is a 2-manifold with boundary, G is a circle-with-chords, and J is an arc. The surfaces satisfy an embedding condition called laundry which is similar to being unknotted. The theorem gives elementary necessary and sufficient conditions for two triples to be equivalent by ambient isotopy. The theorem introduces a new topological invariant called turn. The surfaces of interest can arise from the augmented ribbon model of unknotted single domain protein.