## Experimental Mathematics

### 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.

#### Article information

Source
Experiment. Math., Volume 21, Issue 1 (2012), 26-37.

Dates
First available in Project Euclid: 31 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.em/1338430811

Mathematical Reviews number (MathSciNet)
MR2904905

Zentralblatt MATH identifier
1241.57002

#### Citation

Chas, Moira; Phillips, Anthony. Self-Intersection Numbers of Curves in the Doubly Punctured Plane. Experiment. Math. 21 (2012), no. 1, 26--37. https://projecteuclid.org/euclid.em/1338430811