Open Access
2012 Self-Intersection Numbers of Curves in the Doubly Punctured Plane
Moira Chas, Anthony Phillips
Experiment. Math. 21(1): 26-37 (2012).


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.


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Moira Chas. Anthony Phillips. "Self-Intersection Numbers of Curves in the Doubly Punctured Plane." Experiment. Math. 21 (1) 26 - 37, 2012.


Published: 2012
First available in Project Euclid: 31 May 2012

zbMATH: 1241.57002
MathSciNet: MR2904905

Primary: 57M05
Secondary: 30F99 , 57N50

Keywords: combinatorial length , Doubly punctured plane , free homotopy classes of curves , pair of pants , Self-intersection , thrice-punctured sphere

Rights: Copyright © 2012 A K Peters, Ltd.

Vol.21 • No. 1 • 2012
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