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Seifert Klein bottles for knots with common boundary slopes

Luis G Valdez-Sanchez

Geometry & Topology Monographs 7 (2004) 27–68

DOI: 10.2140/gtm.2004.7.27


We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S3 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 π1–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-π1–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.


Seifert Klein bottles, knot complements, boundary slope

Mathematical Subject Classification

Primary: 57M25

Secondary: 57N10


Received: 10 November 2003
Revised: 10 March 2004
Accepted: 10 March 2004
Published: 17 September 2004

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