  The minimum number of self-intersection points for members of a free homotopy
class of curves on the punctured torus is bounded above in terms of the number
L of letters required for a minimal description of the class in terms of the
generators of the fundamental group and their inverses: it is less than or
equal to (L-2)^2/4 if L is even, and (L-1)(L-3)/4 if L is odd. The classes
attaining this bound are 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 totals are given for classes with self-intersection number
equal to one less than the maximum.
  Proofs use both combinatorial calculations and topological operations on
representative curves. Computer-generated data are tabulated counting, for each
non-negative integer, how many length-L classes have that self-intersection
number, for each length L less than or equal to 12. Experimental data are also
presented for the pair-of-pants surface.
