Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S2, assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A′,B′) are equivalent if there is a homeomorphism from S2 to itself sending A to A′ and B to B′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A′,B′) are equivalent and C is a component code for (A,B), then C is a component code for (A′,B′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C′ are component codes for SCC-pairs (A,B) and (A′,B′), respectively, then (A,B) and (A′,B′) are equivalent if and only if C and C′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.