Closed curves and geodesics with two self-intersections on the punctured torus
This paper is an outgrowth of an NSF REU project at Oregon State University. In Sections 1 and 2, the topological, geometric and number theoretic background for this project is given. In Section 3, the classification up to homeomorphism of the free homotopy classes of closed curves on a once-punctured torus \(T\), whose self-intersection number is two, is obtained. There are eight such classes, six primitive and two imprimitive. The case analysis involved uses standard combinatorial topological methods. The proof that the classes obtained are distinct is done in Section 6. The topological and geometric consequences of the classification are given in Sections 4 and 5. Thus, up to homeomorphism, there is a unique curve in each class realizing the minimum self-intersection number. Given a hyperbolic metric on \(T\), the classification of geodesics on \(T\) with two self-intersections is obtained. Some new results on the Markov spectrum of diophantine approximation are derived in Section 7. Finally, some questions and suggestions for further research are pointed out.
scientific article; zbMATH DE number 1154520
scientific article