On the knot complement problem for non-hyperbolic knots

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Abstract

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.

MSC

57M25
57N10
57M15
57M99

Keywords

3-Manifold
Cosmetic knot
Dehn surgery
Lens space
Satellite knot
Seifert fibered knot

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