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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)
##### Keywords:
width; thin position; satellite knot; connected-sum
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##### References:
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