Thin presentation of knots and lens spaces

Deruelle, A¹ ; Matignon, D¹

¹Université D’Aix-Marseille I, C.M.I. 39, rue Joliot Curie, Marseille Cedex 13, France

Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 677-707

Résumé

This paper concerns thin presentations of knots K in closed 3–manifolds M3 which produce S3 by Dehn surgery, for some slope γ. If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M) with only local maxima, then we show that K is a 0–bridge or 1–bridge braid in M; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally, if the core of the surgery Kγ yields S3 by r–Dehn surgery, then we prove the following inequality: |r|≤ 2g, where g is the genus of Kγ.

Détail

DOI : 10.2140/agt.2003.3.677

Keywords: Dehn surgery, lens space, thin presentation of knots, spines of 3–manifolds

Bibliographie

[1] D Bachman, Non-parallel essential surfaces in knots complements, preprint

[2] J Bailey, D Rolfsen, An unexpected surgery construction of a lens space, Pacific J. Math. 71 (1977) 295

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

[4] J Berge, The knots in $D^2\times S^1$ which have nontrivial Dehn surgeries that yield $D^2\times S^1$, Topology Appl. 38 (1991) 1

[5] R H Bing, An alternative proof that 3–manifolds can be triangulated, Ann. of Math. $(2)$ 69 (1959) 37

[6] S A Bleiler, R A Litherland, Lens spaces and Dehn surgery, Proc. Amer. Math. Soc. 107 (1989) 1127

[7] J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma_{4}=0)$, Lecture Notes in Mathematics 53, Springer (1968)

[8] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $(2)$ 125 (1987) 237

[9] A Deruelle, Conjecture de $\mathbb{RP}^3$ et n\oeuds minimalemant tressés, preprint

[10] M Domergue, Dehn surgery on a knot and real 3–projective space, from: "Progress in knot theory and related topics", Travaux en Cours 56, Hermann (1997) 3

[11] M Domergue, D Matignon, Minimising the boundaries of punctured projective planes in $S^3$, J. Knot Theory Ramifications 10 (2001) 415

[12] M Eudave-Muñoz, Incompressible surfaces in tunnel number one knot complements, Topology Appl. 98 (1999) 167

[13] M Eudave-Muñoz, Incompressible surfaces and $(1,1)$–knots, J. Knot Theory Ramifications 15 (2006) 935

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

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

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

[17] H Goda, M Teragaito, Dehn surgeries on knots which yield lens spaces and genera of knots, Math. Proc. Cambridge Philos. Soc. 129 (2000) 501

[18] C M Gordon, Combinatorial methods in Dehn surgery, from: "Lectures at KNOTS '96 (Tokyo)", Ser. Knots Everything 15, World Sci. Publ., River Edge, NJ (1997) 263

[19] C M Gordon, Dehn surgery on knots, from: "Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990)", Math. Soc. Japan (1991) 631

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

[21] C M Gordon, R A Litherland, Incompressible planar surfaces in 3–manifolds, Topology Appl. 18 (1984) 121

[22] C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371

[23] C M Gordon, J Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996) 385

[24] D J Heath, T Kobayashi, Essential tangle decomposition from thin position of a link, Pacific J. Math. 179 (1997) 101

[25] J A Hoffman, D Matignon, Producing essential 2–spheres, Topology Appl. 124 (2002) 435

[26] K Ichihara, Exceptional surgeries and genera of knots, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 66

[27] W B R Lickorish, A representation of orientable combinatorial 3–manifolds, Ann. of Math. $(2)$ 76 (1962) 531

[28] J Luecke, Dehn surgery on knots in the 3–sphere, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)", Birkhäuser (1995) 585

[29] S V Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990) 101

[30] D Matignon, N Sayari, Longitudinal slope and Dehn fillings, Hiroshima Math. J. 33 (2003) 127

[31] E E Moise, Affine structures in 3–manifolds III: Tubular neighborhoods of linear graphs, Ann. of Math. $(2)$ 55 (1952) 203

[32] L Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971) 737

[33] L P Neuwirth, Knot groups, Annals of Mathematics Studies 56, Princeton University Press (1965)

[34] W Parry, All types implies torsion, Proc. Amer. Math. Soc. 110 (1990) 871

[35] Y Rieck, E Sedgwick, Thin position for a connected sum of small knots, Algebr. Geom. Topol. 2 (2002) 297

[36] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1976)

[37] M Scharlemann, A Thompson, Heegaard splittings of $(\mathrm{surface})\times I$ are standard, Math. Ann. 295 (1993) 549

[38] M Teragaito, Cyclic surgery on genus one knots, Osaka J. Math. 34 (1997) 145

[39] A Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994) 613

[40] A Thompson, Thin position and bridge number for knots in the 3–sphere, Topology 36 (1997) 505

[41] A H Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503

[42] S C Wang, Cyclic surgery on knots, Proc. Amer. Math. Soc. 107 (1989) 1091

[43] Y Q Wu, Cyclic surgery and satellite knots, Topology Appl. 36 (1990) 205

[44] Y Q Wu, The reducibility of surgered 3–manifolds, Topology Appl. 43 (1992) 213

[45] Y Q Wu, Thin position and essential planar surfaces, Proc. Amer. Math. Soc. 132 (2004) 3417