Self-Intersection Numbers of Curves in the Doubly Punctured Plane

We address the problem of computing bounds for the self-intersection number (the minimum number of generic self-intersection points) of members of a free homotopy class of curves in the doubly punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of a set of standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by L-2/4 + L/2 - 1, and that when L is even, this bound is sharp; in that case, there are exactly four distinct classes attaining that bound. For odd L we conjecture a smaller upper bound, (L-2 - 1)/4, and establish it in certain cases in which we show that it is sharp. Furthermore, for the doubly punctured plane, these self-intersection numbers are bounded below, by L/2 - 1 if L is even, and by (L - 1)/2 if L is odd. These bounds are sharp.