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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On simple loops on a solid torus of general genus


Author: Takeshi Kaneto
Journal: Proc. Amer. Math. Soc. 86 (1982), 551-552
MSC: Primary 57N10; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9939-1982-0671234-5
MathSciNet review: 671234
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Abstract: Let $ l$ be a simple loop on the boundary of a solid torus $ T$ of genus $ g$. Let $ {\mathbf{m}}$ be a complete system of oriented meridian disks of $ T$. Let $ W(l,{\mathbf{m}})$ be a word obtained by reading the intersections $ l \cap {\mathbf{m}}$ along $ l$. We shall give a natural method for realizing the cyclic reductions of $ W(l,{\mathbf{m}})$ geometrically. This yields a simple proof of Whitehead-Zieschang's theorem related to the minimality of the intersections of two systems $ \{ {l_1}, \ldots ,{l_n}\} $ and $ {\mathbf{m}}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0671234-5
Keywords: Solid torus of genus $ g$, complete system of meridian disks, simple loops on the boundary, cyclic reductions of words, minimal intersections, pseudominimal Heegaard diagrams
Article copyright: © Copyright 1982 American Mathematical Society