On the knot complement problem for non-hyperbolic knots

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.

The Legendrian knot complement problem

“…Later Matignon [37] proved that all non-hyperbolic knots in atoroidal irreducible Seifert fibered 3-manifolds are determined by their complements (except the cores of the standard Heegaard splittings in Lens spaces).…”

Tangle Equations, the Jones conjecture, and quantum continued fractions

“…Furthermore, Σ(L 1 ) is Seifert, [BZ]. By [Ma,Thm. 1.3], Σ(X) may have two different Dehn fillings yielding (in an orientation preserving way) the same lens space only if Σ(X) is a solid torus and Σ(L 1 ) is a lens space L(p, q) with q 2 = ±1 mod p. The double cover Σ(X) can be a solid torus only if X is rational.…”

Achiral 1-Cusped Hyperbolic 3-Manifolds Not Coming from Amphicheiral Null-homologous Knot Complements

“…By the classification of cosmetic surgeries on a non-hyperbolic knot in a lens space due to Matignon [10], we see that the interior of M n is hyperbolic. In fact, for a non-hyperbolic knot J in a lens space other than S 3 or S 2 × S 1 , if the trivial Dehn surgery and the r-surgery on J are chirally cosmetic, then r = 0, see [10,Theorems 3.2 and 4.1].…”