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\(3\)-manifolds with planar presentations and the width of satellite knots. (English) Zbl 1102.57004
Let \(p:\mathbb{R}^{4}\to\mathbb{R}\) be the standard projection. For the unit three-sphere \(S^{3}\subset\mathbb{R}^{4}\), we regard \(p| _{S^{3}}:{S^{3}}\to\mathbb{R}\) as a height function. Given a knot \(K\) in \(S^{3}\) in general position with respect to \(p\), let \(c_{1}<c_{2}<\dots<c_{n}\) be the critical values of \(h:=p| _{K}\), and \(r_{1}\), \(r_{2},\dots, r_{n-1}\) regular values so that \(c_{i}<r_{i}<c_{i+1}\). Then the width of \(K\) with respect to \(h\), denoted by \(w(K,h)\), is defined to be \(\sum_{i}| K\cap p^{-1}(r_{i})| \). The width of \(K\), denoted by \(w(K)\), is the minimum of \(w(K',h)\) over all knots \(K'\) isotopic to \(K\). A knot \(K\) is in thin position if \(w(K,h)=w(K)\) [cf. D. Gabai, J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)].
In the paper under review, the authors prove that for any satellite knot \(J\) of \(K\), the inequality \(w(J)\geq w(K)\) holds. Since the connected sum \(K_{1}\sharp K_{2}\) is regarded as a satellite of both \(K_{1}\) and \(K_{2}\), the inequality \(w(K_{1}\sharp K_{2})\geq\max\{w(K_{1}),w(K_{2})\}\) holds for any two knots \(K_{1}\) and \(K_{2}\). Compare this with the following results: (1) there exists a knot \(K_{1}\) such that \(w(K_{1}\sharp K_{2})=w(K_{1})\) for any two-bridge knot \(K_{2}\) [M. Scharlemann and A. Thompson, Proceedings of the Casson Fest. Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 135–144 (2004; Zbl 1207.57016)], and (2) if \(K_{1}\) is the connected-sum of meridionally small knots, then for any non-trivial knot \(K_{2}\) the inequality \(w(K_{1}\sharp K_{2})>w(K_{1})\) holds [J. Hendricks, Algebr. Geom. Topol. 4, 1041–1044 (2004; Zbl 1078.57008)]. Here a knot is called meridionally planar small if there is no meridional essential planar surface in its complement.
It is also noted that an obvious upper bound \(w(K_{1}\sharp K_{2})\leq w(K_{1})+w(K_{2})-2\) holds for any two knots \(K_{1}\) and \(K_{2}\), and that the equality holds when they are meridionally planar small knots [Y. Rieck and E. Sedgwick, Algebr. Geom. Topol. 2, 297–309 (2002; Zbl 0991.57004)].
The authors prove the main result on satellite knots in the following way: Suppose that a knot \(K\) is in an unknotted solid torus, and that \(f: W\to S^{3}\) is a knotted embedding. Put \(K'=f(K)\). Then they prove that there is a level-preserving reimbedding \(g\) of \(H:=f(W)\) so that \(g(H)\) is unknotted and untwisted. Therefore \(g(K')\) is isotopic to \(K\), proving \(w(K)\leq w(K')\).

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:
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