Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 2, pages 27--68.
Seifert Klein bottles for knots with common boundary slopes
Luis G Valdez-Sanchez
Abstract.
We consider the question of how many essential Seifert Klein bottles
with common boundary slope a knot in S^3 can bound, up to ambient
isotopy. We prove that any hyperbolic knot in S^3 bounds at most six
Seifert Klein bottles with a given boundary slope. The Seifert Klein
bottles in a minimal projection of hyperbolic pretzel knots of length
3 are shown to be unique and pi_1-injective, with surgery along their
boundary slope producing irreducible toroidal manifolds. The cable
knots which bound essential Seifert Klein bottles are classified;
their Seifert Klein bottles are shown to be non-pi_1-injective, and
unique in the case of torus knots. For satellite knots we show that,
in general, there is no upper bound for the number of distinct Seifert
Klein bottles a knot can bound.
Keywords.
Seifert Klein bottles, knot complements, boundary slope
AMS subject classification.
Primary: 57M25.
Secondary: 57N10.
E-print: arXiv:math.GT/0409459
Submitted to GT on 10 November 2003.
(Revised 10 March 2004.)
Paper accepted 10 March 2004.
Paper published 17 September 2004.
Notes on file formats
Luis G Valdez-Sanchez
Department of Mathematical Sciences, University of Texas at El Paso
El Paso, TX 79968, USA
Email: valdez@math.utep.edu
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