L-space surgery and twisting operation

Motegi, Kimihiko¹

¹Department of Mathematics, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1727-1772

Résumé

A knot in the 3–sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, ie a rational homology 3–sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family {Kn}. We give a sufficient condition for {Kn} to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family {Kn} contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.

Détail

DOI : 10.2140/agt.2016.16.1727

Classification : 57M25, 57M27, 57N10
Keywords: L-space surgery, L-space knot, twisting, seiferter, tunnel number

Bibliographie

[1] I Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000) 431

[2] M Aït-Nouh, D Matignon, K Motegi, Geometric types of twisted knots, Ann. Math. Blaise Pascal 13 (2006) 31

[3] K Baker, A Moore, Montesinos knots, Hopf plumbings, and L-space surgeries, preprint (2014)

[4] R Benedetti, C Petronio, Lectures on hyperbolic geometry, Springer (1992)

[5] J Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript

[6] J Berge, The knots in D2 × S1 which have nontrivial Dehn surgeries that yield D2 × S1, Topology Appl. 38 (1991) 1

[7] M Boileau, J Porti, Geometrization of 3–orbifolds of cyclic type, 272, Société Mathématique de France (2001) 208

[8] S Boyer, C M Gordon, L Watson, On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013) 1213

[9] S Boyer, D Rolfsen, B Wiest, Orderable 3–manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005) 243

[10] J C Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435

[11] A Deruelle, M Eudave-Muñoz, K Miyazaki, K Motegi, Networking Seifert surgeries on knots, IV : Seiferters and branched coverings, from: "Geometry and topology down under" (editors C D Hodgson, W H Jaco, M G Scharlemann, S Tillmann), Contemp. Math. 597, Amer. Math. Soc. (2013) 235

[12] A Deruelle, K Miyazaki, K Motegi, Networking Seifert surgeries on knots, II : The Berge’s lens surgeries, Topology Appl. 156 (2009) 1083

[13] A Deruelle, K Miyazaki, K Motegi, Networking Seifert surgeries on knots, 1021 (2012)

[14] D Eisenbud, U Hirsch, W Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981) 638

[15] M Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, from: "Proceedings of the First Joint Japan–Mexico Meeting in Topology" (editors S García-Ferreira, T Nogura) (2002) 119

[16] M Eudave-Muñoz, E Jasso, K Miyazaki, K Motegi, Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions, Topology Appl. 196 (2015) 729

[17] D Gabai, Foliations and the topology of 3–manifolds, III, J. Differential Geom. 26 (1987) 479

[18] D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1

[19] P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151

[20] C M Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983) 687

[21] C M Gordon, J Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417

[22] C M Gordon, Y Q Wu, Annular Dehn fillings, Comment. Math. Helv. 75 (2000) 430

[23] J E Greene, L-space surgeries, genus bounds, and the cabling conjecture, J. Differential Geom. 100 (2015) 491

[24] A Hatcher, Notes on basic 3–manifold topology, unfinished book manuscript (2007)

[25] M Hedden, On knot Floer homology and cabling, II, Int. Math. Res. Not. 2009 (2009) 2248

[26] J Hom, T Lidman, F Vafaee, Berge–Gabai knots and L-space satellite operations, Algebr. Geom. Topol. 14 (2014) 3745

[27] W Jaco, Lectures on three-manifold topology, 43, Amer. Math. Soc. (1980)

[28] M Jankins, W D Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann. 271 (1985) 381

[29] A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299

[30] M Kouno, K Motegi, T Shibuya, Twisting and knot types, J. Math. Soc. Japan 44 (1992) 199

[31] D T Krcatovich, The reduced knot Floer complex, PhD thesis, Michigan State University (2014)

[32] M Lackenby, R Meyerhoff, The maximal number of exceptional Dehn surgeries, Invent. Math. 191 (2013) 341

[33] T Lidman, A H Moore, Pretzel knots with L-space surgeries, Michigan Math. J. 65 (2016) 105

[34] P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact 3–manifolds, III, J. Symplectic Geom. 5 (2007) 357

[35] Y Mathieu, Unknotting, knotting by twists on disks and property (P) for knots in S3, from: "Knots 90" (editor A Kawauchi), de Gruyter (1992) 93

[36] K Miyazaki, K Motegi, Seifert fibred manifolds and Dehn surgery, Topology 36 (1997) 579

[37] K Miyazaki, K Motegi, Seifert fibered manifolds and Dehn surgery, III, Comm. Anal. Geom. 7 (1999) 551

[38] J M Montesinos, Seifert manifolds that are ramified two-sheeted cyclic coverings, Bol. Soc. Mat. Mexicana 18 (1973) 1

[39] J M Montesinos, Surgery on links and double branched covers of S3, from: "Knots, groups, and 3–manifolds (Papers dedicated to the memory of R H Fox)" (editor L P Neuwirth), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 227

[40] K Morimoto, M Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991) 143

[41] K Motegi, K Tohki, On L-space twisted torus knots, in preparation

[42] R Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv. 69 (1994) 155

[43] Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577

[44] Y Ni, Erratum : Knot Floer homology detects fibred knots, Invent. Math. 177 (2009) 235

[45] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311

[46] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159

[47] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281

[48] P S Ozsváth, Z Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1

[49] C Petronio, J Porti, Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem, Expo. Math. 18 (2000) 1

[50] Z Szabó, Lecture notes on Heegaard Floer homology, from: "Low dimensional topology" (editors T S Mrowka, P S Ozsváth), IAS/Park City Math. Ser. 15, Amer. Math. Soc. (2009) 197

[51] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[52] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357

[53] F Vafaee, On the knot Floer homology of twisted torus knots, Int. Math. Res. Not. 2015 (2015) 6516