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.

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Luis G Valdez-Sanchez
Department of Mathematical Sciences, University of Texas at El Paso
El Paso, TX 79968, USA

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