Tangles and tubing operations
Let \(K\) be a knot in \(S^3\), and \(E(K)\) the exterior \(S^3-\text{int }N(K)\). We denote the 3-manifold obtained by a Dehn surgery on \(K\) along a slope \(\gamma\) by \(K(\gamma)\). A 3-manifold is laminar if it contains an essential lamination. The essential lamination is a generalization of the essential surface as well as the taut foliation. \textit{D. Gabai} and \textit{U. Oertel} showed [Ann. Math., II. Ser. 130, No. 1, 41-73 (1989; Zbl 0685.57007)] that the universal covering of a laminar 3-manifold is the \(\mathbb{R}^3\). In [J. Differ. Geom. 43, No. 1, 171-197 (1996; Zbl 0851.57018)] \textit{Y.-Q. Wu} showed that \(K(\gamma)\) is laminar for an arbitrary nonmeridional slope \(\gamma\), if \(K\) is an arborescent knot and not a Montesinos knot of length at most 3. Hence these knots have property \(P\) and satisfy the cabling conjecture. Remember that every nonintegral Dehn surgery on a knot \(K\) yields a Haken manifold if \(K\) is a sum of two nonsplit tangles as shown in Theorem 2.0.3 in [\textit{M. Culler, C. McA. Gordon, J. Luecke}, and \textit{P. B. Shalen}, Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006)]. The main techniques Wu used are decompositions of knots into two nonsplit 2-string tangles and tubing operations. In this paper, tangles with more strings are considered. Let \((B,T)\) be an \(n\)-string tangle, \(E(T)\) the exterior \(\text{cl}(B-N(T))\) and \(P\) the punctured sphere \(\text{cl}(\partial B-N(T))\). 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 \(T(s)=P\cup (E(T)\cap N(s))\) is said to be obtained by performing a tubing operation on \(P\) along \(s\). The author gives a necessary and sufficient condition for \(T(s)\) to be incompressible in \(E(T)\), when \((B,T)\) is atomic. The author shows 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.
scientific article; zbMATH DE number 1307872