Once-punctured Klein bottles in knot complements
For a knot \(K\) in the \(3\)-sphere \(S^3\), a Seifert Klein bottle of \(K\) is a once-punctured Klein bottle \(P\) with \(\partial P=K\). If \(K\) is non-trivial and not a \(2\)-cable, then \(F\) is essential. That is, the restriction of \(F\) to the knot exterior \(E(K)\) is incompressible and boundary-incompressible. Furthermore, it was shown by \textit{K.\ Ichihara, M.\ Ohtouge} and \textit{M.\ Teragaito} [Topology Appl. 122, No. 3, 467--478 (2002; Zbl 0997.57009)] that if \(K\) admits two Seifert Klein bottles with different boundary slopes then \(K\) admits just two such slopes and their slopes are at a distance \(4\) or \(8\) from each other, and if at distance \(8\) then \(K\) is the figure-eight knot. Also, they conjectured that if at distance \(4\) then \(K\) must belong to a certain family of knots. The paper under review gives an affirmative solution to this conjecture. As a corollary, the authors obtain that the figure-eight knot and the \((-2,3,7)\)-pretzel knot (and its mirror image) are the only hyperbolic knots with more than one Seifert Klein bottle boundary slope. The argument is the analysis of the graph pair coming from intersection between two Seifert Klein bottles. Also, there is a discussion in a general situation. For an irreducible \(3\)-manifold \(M\) with an incompressible torus boundary component \(T\), consider the case that \(M\) contains two essential once-punctured Klein bottles whose boundaries lie on \(T\). Then their boundary slopes are shown to be at a distance at most \(8\). Moreover, if the distance is \(6\) or \(8\), then the manifold \(M\) is determined.
scientific article; zbMATH DE number 2136462