Self-Intersection Numbers of Curves in the Doubly Punctured Plane

Abstract

We address the problem of computing bounds for the selfintersection 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.