✦ LIBER ✦

Tangles and tubing operations

✍ Scribed by Chuichiro Hayashi

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 613 KB
Volume: 92
Category: Article
ISSN: 0166-8641
DOI: 10.1016/s0166-8641(97)00250-2

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✦ Synopsis

Let (B, T) be an n string tangle, E(T) the exterior cl(B --N(T)) and P the punctured sphere cl(aB -N(T)). The tangle (B, T) is called atomic if it does not contain a nonsplit tangle with k < n essentially. For a string s of T the surface T(s) = PU (E(T) nN( s)) is said to be obtained by performing a tubing operation on P along s. We give a necessary and sufficient condition for T(s) to be incompressible in E(T), when (B,T) is atomic. We show also that if a knot K is decomposed into two atomic tangles with no parallel pairs of strings, then every nontrivial Dehn surgery on K yields a laminar manifold.

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