##
**\(3\)-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 \(w(K)\leq w(K')\).

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 \(w(K)\leq w(K')\).

Reviewer: Hitoshi Murakami (Tokyo)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

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\textit{M. Scharlemann} and \textit{J. Schultens}, Trans. Am. Math. Soc. 358, No. 9, 3781--3805 (2006; Zbl 1102.57004)

### References:

[1] | Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. · Zbl 0568.57001 |

[2] | Ralph H. Fox, On the imbedding of polyhedra in 3-space, Ann. of Math. (2) 49 (1948), 462 – 470. · Zbl 0032.12502 |

[3] | David Gabai, Foliations and the topology of 3-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461 – 478. David Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479 – 536. · Zbl 0627.57012 |

[4] | W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. · Zbl 0886.57001 |

[5] | Kanji Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3527 – 3532. · Zbl 0854.57006 |

[6] | Kanji Morimoto and Jennifer Schultens, Tunnel numbers of small knots do not go down under connected sum, Proc. Amer. Math. Soc. 128 (2000), no. 1, 269 – 278. · Zbl 0928.57003 |

[7] | Yo’av Rieck and Eric Sedgwick, Thin position for a connected sum of small knots, Algebr. Geom. Topol. 2 (2002), 297 – 309. · Zbl 0991.57004 |

[8] | Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. · Zbl 0339.55004 |

[9] | Martin Scharlemann, Handlebody complements in the 3-sphere: a remark on a theorem of Fox, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1115 – 1117. · Zbl 0759.57012 |

[10] | Martin Scharlemann and Jennifer Schultens, Annuli in generalized Heegaard splittings and degeneration of tunnel number, Math. Ann. 317 (2000), no. 4, 783 – 820. · Zbl 0953.57002 |

[11] | M. Scharlemann, A. Thompson, On the additivity of knot width, ArXiv preprint math.GT/0403326. · Zbl 1207.57016 |

[12] | Horst Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245 – 288 (German). · Zbl 0058.17403 |

[13] | Jennifer Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 3, 539 – 544. · Zbl 1054.57011 |

[14] | A. Thompson, personal communication. |

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