Markoff spectrum, diophantine approximation
Springer Online Journal Archives 1860-2000
Abstract 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 hyperbolicT 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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