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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Research started during the Summer 1994 NSF REU Program at Oregon State University and partially supported by NSF DMS 9300281
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Crisp, D., Dziadosz, S., Garity, D.J. et al. Closed curves and geodesics with two self-intersections on the Punctured torus. Monatshefte für Mathematik 125, 189–209 (1998). https://doi.org/10.1007/BF01317313
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DOI: https://doi.org/10.1007/BF01317313