Seifert Klein bottles for knots with common boundary slopes
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_1injective, 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 nonpi_1injective, 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2004
 arXiv:
 arXiv:math/0409459
 Bibcode:
 2004math......9459V
 Keywords:

 Mathematics  Geometric Topology;
 57M25;
 57N10
 EPrint:
 Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper2.abs.html