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.