CLOSED CURVES AND GEODESICS WITH 2 SELF-INTERSECTIONS ON THE PUNCTURED TORUS

We classify the free homotopy classes of closed curves with minimal se lf intersection number two on a once punctured torus, T, up to homeomo rphism. Of these, there are six primitive classes and two imprimitive. The classification leads to the topological result that, up to homeom orphism, there is a unique curve in each class realizing the minimum s elf intersection number. The classification yields a complete classifi cation of geodesics on hyperbolic T which have self intersection numbe r two. We also derive new results on the Markoff spectrum of diophanti ne approximation; in particular, exactly three of the imprimitive clas ses correspond to families of Markoff values below Hall's ray.