A joint paper with S. Dziadosz and P. Wiles, presented at the January 1995 AMS-MAA meeting (download a copy). A revised version was subsequently published as:
Crisp, David; Dziadosz, Susan; Garity, Dennis J.; Insel, Thomas; Schmidt, Thomas A.; Wiles, Peter. Closed curves and geodesics with two self-intersections on the punctured torus. Monatsh. Math. 125 (1998), no. 3, 189—209.
We classify the free homotopy classes of closed curves with minimal self intersection number two on a once punctured torus, T, up to homeomorphism. Of these, there are six primitive classes and two imprimitive. The classification leads to the topological result that, up to homeomorphism, there is a unique curve in each class realizing the minimum self intersection number. The classification yields a complete classification of geodesics on hyperbolic T which have self intersection number two. We also derive new results on the Markoff spectrum of diophantine approximation; in particular, exactly three of the imprimitive classes correspond to families of Markoff values below Hall’s ray.
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