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All integral slopes can be Seifert fibered slopes for hyperbolic knots. (English) Zbl 1083.57012

For a hyperbolic knot in the \(3\)-sphere, all but finitely many slopes yield hyperbolic manifolds by Thurston’s hyperbolic Dehn surgery theorem. Exceptional surgeries are split into four cases, according to the resulting manifold: a reducible manifold, a Seifert fiber space, a toroidal manifold or a counterexample to the geometrization conjecture. The first and last ones are expected not to happen. Also, it is conjectured that a Seifert fiber space is only obtained by an integral slope.
The paper under review shows that for each integer \(n\), there exists a tunnel number one hyperbolic knot on which \(n\)-surgery yields a Seifert fiber space over the \(2\)-sphere with three exceptional fibers. The analogous result for toroidal surgery was already known by M. Teragaito [Proc. Am. Math. Soc. 130, No. 9, 2803–2808 (2002; Zbl 0997.57033)]. The proof is constructive, and uses the Montesinos trick to show that the resulting manifold is a Seifert fiber space. Finally, it is verified that there exists a trivial knot disjoint from the knot which becomes a Seifert fiber in the resulting Seifert fiber space. This supports a conjecture by K. Miyazaki and K. Motegi [Commun. Anal. Geom. 7, No. 3, 551–583 (1999; Zbl 0940.57025)].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds

Software:

SnapPea
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References:

[1] M Aït Nouh, D Matignon, K Motegi, Twisted unknots, C. R. Math. Acad. Sci. Paris 337 (2003) 321 · Zbl 1033.57003
[2] M Aït Nouh, D Matignon, K Motegi, Obtaining graph knots by twisting unknots, Topology Appl. 146/147 (2005) 105 · Zbl 1086.57008
[3] M Aït-Nouh, D Matignon, K Motegi, Geometric types of twisted knots, Ann. Math. Blaise Pascal 13 (2006) 31 · Zbl 1158.57005
[4] J Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript
[5] S A Bleiler, Knots prime on many strings, Trans. Amer. Math. Soc. 282 (1984) 385 · Zbl 0545.57001
[6] S A Bleiler, Prime tangles and composite knots, Lecture Notes in Math. 1144, Springer (1985) 1 · Zbl 0596.57003
[7] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. \((2)\) 125 (1987) 237 · Zbl 0633.57006
[8] M Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries (2002) 119 · Zbl 1009.57010
[9] R Fintushel, R J Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980) 33 · Zbl 0425.57001
[10] D Gabai, Foliations and the topology of 3-manifolds III, J. Differential Geom. 26 (1987) 479 · Zbl 0639.57008
[11] F González-Acuña, H Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986) 89 · Zbl 0591.57002
[12] C M Gordon, Dehn filling, New Stud. Adv. Math. 3, Int. Press, Somerville, MA (2003) 41 · Zbl 1044.57005
[13] C M Gordon, J Luecke, Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Cambridge Philos. Soc. 102 (1987) 97 · Zbl 0655.57500
[14] C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371 · Zbl 0672.57009
[15] C M Gordon, J Luecke, Dehn surgeries on knots creating essential tori I, Comm. Anal. Geom. 3 (1995) 597 · Zbl 0865.57015
[16] C M Gordon, J Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417 · Zbl 1062.57006
[17] K Ichihara, K Motegi, H J Song, Longitudinal Seifert fibered surgeries on hyperbolic knots, preprint
[18] M Kouno, K Motegi, T Shibuya, Twisting and knot types, J. Math. Soc. Japan 44 (1992) 199 · Zbl 0739.57003
[19] P B Kronheimer, T S Mrowka, Dehn surgery, the fundamental group and SU\((2)\), Math. Res. Lett. 11 (2004) 741 · Zbl 1084.57006
[20] P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. \((2)\) 165 (2007) 457 · Zbl 1204.57038
[21] Y Mathieu, Unknotting, knotting by twists on disks and property \((\mathrmP)\) for knots in \(S^3\), de Gruyter (1992) 93 · Zbl 0772.57012
[22] T Mattman, K Miyazaki, K Motegi, Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions, Trans. Amer. Math. Soc. 358 (2006) 4045 · Zbl 1105.57004
[23] W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37 · Zbl 0525.57003
[24] K Miyazaki, K Motegi, Seifert fibered manifolds and Dehn surgery III, Comm. Anal. Geom. 7 (1999) 551 · Zbl 0940.57025
[25] J M Montesinos, Surgery on links and double branched covers of \(S^3\), Princeton Univ. Press (1975) · Zbl 0325.55004
[26] K Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995) 3527 · Zbl 0854.57006
[27] K Motegi, An experimental study of Seifert fibered Dehn surgery via SnapPea (2003) 95 · Zbl 1052.57002
[28] M Teragaito, Toroidal surgeries on hyperbolic knots, Proc. Amer. Math. Soc. 130 (2002) 2803 · Zbl 0997.57033
[29] W P Thurston, The geometry and topology of 3-manifolds, lecture notes, Princeton University (1979)
[30] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. \((\)N.S.\()\) 6 (1982) 357 · Zbl 0496.57005
[31] J Weeks, SnapPea: a computer program for creating and studying hyperbolic 3-manifolds
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