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.
Keywords
Seifert Klein bottles, knot complements,
boundary slope