Seifert Klein bottles for knots with common boundary slopes

الأصل · EN

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.