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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク satellite knot ： ウィキペディア英語版 satellite knot In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement.〔Colin Adams, ''The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots'', (2001), ISBN 0-7167-4219-5〕 Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots and Whitehead doubles. (''See'' Basic families, below for definitions of the last two classes.) A satellite ''link'' is one that orbits a companion knot ''K'' in the sense that it lies inside a regular neighborhood of the companion. A satellite knot $K$ can be picturesquely described as follows: start by taking a nontrivial knot $K\text{'}$ lying inside an unknotted solid torus $V$. Here "nontrivial" means that the knot $K\text{'}$ is not allowed to sit inside of a 3-ball in $V$ and $K\text{'}$ is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot. This means there is a non-trivial embedding $f\backslash colon\; V\; \backslash to\; S^3$ and $K=f(K\text{'})$. The central core curve of the solid torus $V$ is sent to a knot $H$, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" $K$ orbits.The construction ensures that $f(\backslash partial\; V)$ is a non-boundary parallel incompressible torus in the complement of $K$. Composite knots contain a certain kind of incompressible torus called a swallow-follow torus, which can be visualized as swallowing one summand and following another summand. Since $V$ is an unknotted solid torus, $S^3\; \backslash setminus\; V$ is a tubular neighbourhood of an unknot $J$. The 2-component link $K\text{'}\; \backslash cup\; J$ together with the embedding $f$ is called the ''pattern'' associated to the satellite operation. A convention: people usually demand that the embedding $f\; \backslash colon\; V\; \backslash to\; S^3$ is ''untwisted'' in the sense that $f$ must send the standard longitude of $V$ to the standard longitude of $f(V)$. Said another way, given two disjoint curves $c\_1,c\_2\; \backslash subset\; V$, $f$ must preserve their linking numbers i.e.: $lk(f(c\_1),f(c\_2))=lk(c\_1,c\_2)$. ==Basic families== When $K\text{'}\; \backslash subset\; \backslash partial\; V$ is a torus knot, then $K$ is called a ''cable knot.'' Examples 3 and 4 are cable knots. If $K\text{'}$ is a non-trivial knot in $S^3$ and if a compressing disc for $V$ intersects $K\text{'}$ in precisely one point, then $K$ is called a ''connect-sum.'' Another way to say this is that the pattern $K\text{'}\; \backslash cup\; J$ is the connect-sum of a non-trivial knot $K\text{'}$ with a Hopf link. If the link $K\text{'}\; \backslash cup\; J$ is the Whitehead link, $K$ is called a ''Whitehead double.'' If $f$ is untwisted, $K$ is called an untwisted Whitehead double. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「satellite knot」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.