On the punctured torus the number of essential self-intersections of a homotopy class of closed curves is bounded sharply by a quadratic function of its combinatorial length the number of letters required for its minimal description in terms of the two generators of the fundamental group and their inverses.
Similar descriptions and counts are given for classes with self-intersection number equal to one less than the bound. Proofs use both combinatorial calculations and topological operations on representative curves. Such experiments led to the results above. Experimental data are also presented for the pair-of-pants surface.
Curves and numbers
Source Experiment. Zentralblatt MATH identifier Keywords Punctured torus free homotopy classes of curves self-intersection. Chas, Moira; Phillips, Anthony.
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