Nœuds qui ne sont pas déterminés par leur complément dans les 3-variétés à bord

Domergue, Michel; Mathieu, Yves

doi:10.24033/bsmf.2170

Bul. Soc. Math. France

1991

DOI: 10.24033/bsmf.2170

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Nœuds qui ne sont pas déterminés par leur complément dans les 3-variétés à bord

Michel Domergue¹,

Yves Mathieu²

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On the knot complement problem for non-hyperbolic knots

Matignon

2010

Topology and its Applications

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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.

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On the knot complement problem for non-hyperbolic knots

Matignon

2010

Topology and its Applications

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DISTANCES BETWEEN THE SURGERY SLOPES YIELDING S³ ON A GENERIC LINK

Mayrand

2004

J. Knot Theory Ramifications

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A generic link is a 2-component link L=(k1,k2) in S3, such that none of the components is unknotted, and such that the complement of the link contains no annulus whose boundary is the union of an essential circle on ∂N(k1) and an essential circle on ∂N(k2). Let μ1 and μ2 be the meridian slopes of k1 and k2 respectively. If the non-trivial (β1,β2)-surgery on L yields S3, then by the complement theorem, β1≠μ1 and β2≠μ2. We find a bound for min (Δ(β1,μ1),Δ(μ2,β2)), where Δ(μi,βi) is the distance between the slopes μi and βi on ∂N(ki)(i∈{1,2}). The bound depends on the bridge number of L.

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Nœuds qui ne sont pas déterminés par leur complément dans les 3-variétés à bord

Cited by 2 publications

References 1 publication

On the knot complement problem for non-hyperbolic knots

On the knot complement problem for non-hyperbolic knots

DISTANCES BETWEEN THE SURGERY SLOPES YIELDING S³ ON A GENERIC LINK

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Nœuds qui ne sont pas déterminés par leur complément dans les 3-variétés à bord

Cited by 2 publications

References 1 publication

On the knot complement problem for non-hyperbolic knots

On the knot complement problem for non-hyperbolic knots

DISTANCES BETWEEN THE SURGERY SLOPES YIELDING S3 ON A GENERIC LINK

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DISTANCES BETWEEN THE SURGERY SLOPES YIELDING S³ ON A GENERIC LINK