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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Extending closed plane curves to immersions of the disk with $n$ handles


Author: Keith D. Bailey
Journal: Trans. Amer. Math. Soc. 206 (1975), 1-24
MSC: Primary 57D40
DOI: https://doi.org/10.1090/S0002-9947-1975-0370621-3
MathSciNet review: 0370621
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Abstract: Let $f:S \to E$ be a normal curve in the plane. The extensions of $f$ to immersions of the disk with $n$ handles $({T_n})$ can be determined as follows. A word for $f$ is constructed using the definitions of Blank and Marx and a combinatorial structure, called a ${T_n}$-assemblage, is defined for such words. There is an immersion extending $f$ to ${T_n}$ iff the tangent winding number of $f$ is $1 - 2n$ and $f$ has a ${T_n}$-assemblage. For each $n$, a canonical curve ${f_n}$ with a topologically unique extension to ${T_n}$ is described (${f_0}$ = Jordan curve). Any extendible curve with the minimum number $(2n + 2\;{\text {for}}\;n > 0)$ of self-intersections is equivalent to ${f_n}$.


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Article copyright: © Copyright 1975 American Mathematical Society