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 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 L2/4 + 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, (L2 − 1)/4, and establish it in certain cases, where we show it is sharp. Furthermore, for the doubly-punctured plane, these self-intersection numbers are bounded below, by L/2 − 1 if L is even, (L − 1)/2 if L is odd; these bounds are sharp.