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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:
 [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 · doi:10.2307/1969291 · doi.org [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 · doi:10.2140/agt.2002.2.297 · doi.org [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 · doi:10.1007/PL00004423 · doi.org [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 · doi:10.1007/BF01181346 · doi.org [13] Jennifer Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 3, 539 – 544. · Zbl 1054.57011 · doi:10.1017/S0305004103006832 · doi.org [14] A. Thompson, personal communication.
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