Seifert Klein bottles for knots with common boundary slopes

Details Seifert Klein bottles for knots with common boundary slopes Seifert Klein bottles for knots with common boundary slopes

2004-09-23

arxiv.org/abs/math/0409459v2

Geom. Topol. Monogr. 7 (2004) 27-68

Mathematics

Geometric Topology

Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper2.abs.html

Scientific paper

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.

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