Self-Intersection Numbers of Curves in the Doubly Punctured Plane

Abstract

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.

2000 AMS Subject Classification:

• Primary 57M05
• Secondary 57N50
• 30F99

Keywords:

• Doubly punctured plane
• thrice-punctured sphere
• pair of pants
• free homotopy classes of curves
• self-intersection
• combinatorial length

ACKNOWLEDGMENTS

The authors have benefited from discussions with Dennis Sullivan, and are very grateful to Igor Rivin, who contributed an essential element to the proof of Theorem 1.7. Additionally, they have profited from use of Chris Arettines’ JAVA program, which draws minimally self-intersecting representatives of free homotopy classes of curves in surfaces. The program is currently available online. ²

Notes

¹Available at http://www.math.sunysb.edu/moira/applets/intersectionApplet.html.

²At www.math.sunysb.edu/~moira/applets/chrisApplet.html.