On the knot complement problem for non-hyperbolic knots
The paper under review introduces two infinite families of knots in lens spaces. One of them consists of knots whose exteriors are Seifert fibered manifolds, and the other consists of knots whose exteriors contain essential tori. Each knot of those families admits a non-trivial Dehn surgery which creates a lens space homeomorphic to the original lens space, called a cosmetic surgery in the literature, under an orientation-reversing homeomorphism. Such a surgery is shown to be unique. No two knots of each family have homeomorphic exteriors. The main point is that these families give all non-hyperbolic non-trivial knots in lens spaces which admit cosmetic surgeries. The construction of the families is concrete. On the other hand, \textit{S. A. Bleiler, C. D. Hodgson} and \textit{J. R. Weeks} [Geom. Topol. Monogr. 2, 23--34 (1999; Zbl 0948.57017)] gave a single example of a hyperbolic knot in a lens space which admits a cosmetic surgery. The author of the current paper announces an infinite family of such hyperbolic knots. As a corollary, non-hyperbolic knots in lens spaces, more generally, closed atoroidal irreducible Seifert fibered manifolds, are determined by their complements, except for the axes in lens spaces \(L(p,q)\) with \(q^2\equiv \pm 1\pmod{p}\).
scientific article; zbMATH DE number 5724585
scientific article