Self-Intersection Numbers of Curves on the Punctured Torus

Abstract

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). We show that if a homotopy class has combinatorial length $L$, then its number of essential self-intersections is bounded by $(L-2)^{2}/4$ if $L$ is even, and $(L-1)(L-3)/4$ if $L$ is odd. The classes attaining this bound can be explicitly described in terms of the generators; there are $(L-2)^2+ 4$ of them if $L$ is even, and $2(L-1)(L-3)+8$ if $L$ is odd. 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.

Computer-generated data are tabulated by counting for each nonnegative integer how many length-$L$ classes have that self-intersection number, for each length $L$ less than or equal to $13$. Such experiments led to the results above. Experimental data are also presented for the pair-of-pants surface.