Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T12:57:33.309Z Has data issue: false hasContentIssue false

Klein slopes on hyperbolic 3-manifolds

Published online by Cambridge University Press:  01 September 2007

DANIEL MATIGNON
Affiliation:
C.M.I. Université de Provence, 39 rue Joliot Curie, 13453 Marseille, Cedex 13, France. email: [email protected]
NABIL SAYARI
Affiliation:
Département de Mathématiques et de Statistique, Université de Moncton, NB, Canada. email: [email protected]

Abstract

This paper is devoted to 3-manifolds which admit two distinct Dehn fillings producing a Klein bottle.

Let M be a compact, connected and orientable 3-manifold whose boundary contains a 2-torus T. If M is hyperbolic then only finitely many Dehn fillings along T yield non-hyperbolic manifolds. We consider the situation where two distinct slopes γ1, γ2 produce a Klein bottle. We give an upper bound for the distance Δ(γ1, γ2), between γ1 and γ2. We show that there are exactly four hyperbolic manifolds for which Δ(γ1, γ2) > 4.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Gordon, C. McA.. Boundary slopes of punctured tori in 3-manifolds. Trans. Amer. Math. Soc. 350, no 5 (1998), 17131790.CrossRefGoogle Scholar
[2]Gordon, C. McA.. Combinatorial methods in Dehn surgery. Lectures at Knots'96 (World Scientific Publishing Co., 1997) 263290.Google Scholar
[3]Gordon, C. McA.. Dehn filling: a survey. Knot Theory (Warsaw, 1995), (Banach Center Publ.) Polish Acad. Sci. 42 (1998), 129–144.CrossRefGoogle Scholar
[4]Gordon, C. McA.. Small surfaces and Dehn filling. Geom. Topol. Monogr. 2 (1999), Proceedings of the Kirbyfest, 177199.CrossRefGoogle Scholar
[5]Gordon, C. McA. and Litherland, R. A.. Incompressible planar surfaces in 3-manifolds. Topology Appl. 18 (1984), 12144.CrossRefGoogle Scholar
[6]Gordon, C. McA. and Luecke, J.. Dehn surgeries on knots creating essential tori, I. Comm. Anal. Geom. 4 (1995), 597644.CrossRefGoogle Scholar
[7]Ichihara, K. and Teragaito, M.. Klein bottle surgery and genera of knots. Pacific J. Math. 210, no. 2 (2003), 317333.CrossRefGoogle Scholar
[8]Jaco, W.. Lectures on three-manifold topology. 43 Conference board of the mathematical sciences (1980).CrossRefGoogle Scholar
[9]Kuwako, K.. Dehn surgeries creating Klein bottles, private communication.Google Scholar
[10]Lee, S.. Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components, preprint.Google Scholar
[11]Oh, S.. Dehn filling, reducible 3-manifolds and Klein bottles. Proc. Amer. Math. Soc. 126 (1998), 289296.CrossRefGoogle Scholar
[12]Ramírez–Losada, E. and Valdez–Sánchez, L. G.. Once-punctured Klein bottles in knot complements. Topology Appl. 146–147 (2005), 159188.CrossRefGoogle Scholar
[13]Rolfsen, D.. Knots and Links (Publish or Perish, 1976).Google Scholar
[14]Teragaito, M.. Dehn surgeries on composite knots creating Klein bottles. J. Knot Theory Ramifications 8, no 3 (1999), 391395.CrossRefGoogle Scholar
[15]Teragaito, M.. Creating Klein bottles by surgery on knots. J. Knot Theory Ramifications 10, no 5 (2001), 781794.CrossRefGoogle Scholar
[16]Thurston, W. P.. The geometry and topology of 3-manifolds. Lecture notes (Princeton University, 1979).Google Scholar
[17]Thurston, W. P.. Three dimensional manifolds, Klein groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357381.CrossRefGoogle Scholar