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Transactions of the American Mathematical Society

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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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DOI: https://doi.org/10.1090/S0002-9947-1975-0370621-3
Article copyright: © Copyright 1975 American Mathematical Society

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