Networking Seifert surgeries on knots, III

Deruelle, Arnaud¹ ; Miyazaki, Katura² ; Motegi, Kimihiko³

¹Institute of Natural Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan
²Faculty of Engineering, Tokyo Denki University, 5 Senju-Asahi-cho, Adachi-ku, Tokyo 120-8551, Japan
³Department of Mathematics, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan

Algebraic and Geometric Topology, Tome 14 (2014) no. 4, pp. 2065-2101

Résumé

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced by the authors in [Mem. Amer. Math. Soc. 217 (2012), no. 1021]. The Seifert surgery network is a 1–dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a “hyperbolic seiferter” or a “hyperbolic annular pair” produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots that have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted.

Détail

DOI : 10.2140/agt.2014.14.2065

Classification : 57M25, 57M50, 57N10
Keywords: Dehn surgery, hyperbolic knot, Seifert fiber space, seiferter, Seifert surgery network, band-sum

Bibliographie

[1] K Asano, Homeomorphisms of prism manifolds, Yokohama Math. J. 26 (1978) 19

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

[3] J Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript

[4] M Boileau, J Porti, Geometrization of $3$–orbifolds of cyclic type, Astérisque 272 (2001) 208

[5] F Bonahon, J P Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. 16 (1983) 451

[6] 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" (editor C D Hodgson), Amer. Math. Soc. (2013) 235

[7] A Deruelle, K Miyazaki, K Motegi, Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network, preprint

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

[9] A Deruelle, K Miyazaki, K Motegi, Networking Seifert surgeries on knots, Mem. Amer. Math. Soc. 1021, Amer. Math. Soc. (2012)

[10] M Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35

[11] M Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, from: "Proceedings of the First Joint Japan–Mexico Meeting in Topology", Topology Appl. 121 (2002) 119

[12] R Fintushel, R J Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980) 33

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

[14] J E Greene, $L$–space surgeries, genus bound, and the cabling conjecture, preprint (2012)

[15] A E Hatcher, Notes on basic $3$–manifold topology, lecture notes

[16] M Hirasawa, K Shimokawa, Dehn surgeries on strongly invertible knots which yield lens spaces, Proc. Amer. Math. Soc. 128 (2000) 3445

[17] C Hodgson, J H Rubinstein, Involutions and isotopies of lens spaces, from: "Knot theory and manifolds" (editor D Rolfsen), Lecture Notes in Math. 1144, Springer (1985) 60

[18] K Ishihara, K Motegi, Band sum operations yielding trivial knots, Bol. Soc. Mat. Mexicana 15 (2009) 103

[19] W Jaco, Lectures on three–manifold topology, CBMS Reg. Conf. Series Math. 43, Amer. Math. Soc. (1980)

[20] B Jiang, S Wang, Y Q Wu, Homeomorphisms of 3–manifolds and the realization of Nielsen number, Comm. Anal. Geom. 9 (2001) 825

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

[22] W D Neumann, F Raymond, Seifert manifolds, plumbing, $\mu $–invariant and orientation reversing maps, from: "Algebraic and geometric topology" (editor K C Millett), Lecture Notes in Math. 664, Springer (1978) 163

[23] U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209

[24] P Orlik, Seifert manifolds, Lecture Notes in Mathematics 291, Springer (1972)

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

[26] J H Rubinstein, On $3$–manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979) 129

[27] M Scharlemann, A Thompson, Link genus and the Conway moves, Comment. Math. Helv. 64 (1989) 527

[28] W P Thurston, The geometry and topology of $3$–manifolds, lecture notes, Princeton University (1979)

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