Protein folding invariants

This article discusses the topological invariants associated with an augmented ribbon model of single domain protein. The model is a triple (S, G, J) in S3 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. The invariants are necessary and sufficient conditions for two triples to be equivalent by ambient isotopy. The model describes the native state, the unfolded state, and a unique folding pathway as a single mathematical entity. This may help illuminate some of the remarkable properties of protein. Twist transitions are introduced that allow the surface to pass through itself. A new arithmetic involving the complex numbers is used to represent variable linking numbers.