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 π 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-π 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.

Toroidal and Klein bottle boundary slopes

“…Following ∂ 1 S around in Fig. 15, we can see that ∂ 1 S = (yx) 2 y t−2 , which is not a primitive word in π 1 (S B ) = {x, y | −}; therefore S is incompressible in S B by [4,Lemma 5.2]. Similarly, in S W we can take as complete disk system the white bigon X in G T,S with corners along v 3 and v 4 and the white t + 4-sided face Y containing e t+2 ; we then compute that ∂ 1 S ⊂ ∂S W is represented by the word Y t+3 XY X in π 1 (S W ) = {X, Y | −}, which is not primitive, so S is incompressible in S W too.…”