2004/09/25 by Luis G. Valdez-Sanchez
Mathematics · #math.GT #msc:57M25 #msc:57N10
published as Geom. Topol. Monogr. 7 (2004) 27-68 · Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper2.abs.html
arxiv created 2004/09/25 · arxiv updated 2009/12/01
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 pi1-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-pi1-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.