Experimental Mathematics

Self-Intersection Numbers of Curves in the Doubly Punctured Plane

Moira Chas and Anthony Phillips

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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

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

First available in Project Euclid: 31 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 57N50: $S^{n-1}\subset E^n$, Schoenflies problem 30F99: None of the above, but in this section

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


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

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