Narrow Results

It is proved by Ozawa that every knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle sphere. In this paper, we show that for every integer $n$ with $n\ge 2$, there is a knot in the 3-sphere which admits an essential free $n$-tangle decomposition and also admits at least $2n-3$ distinct essential tangle spheres. To this end, we give examples of nested essential free $n$-tangles for every integer $n$ with $n\ge 3$.

In this paper, we show that free genus one knots do not admit essential tangle decompositions.

We prove that if K⊂ S³ is either: (I) a link with an essential n-string arcbody decomposition, where at least one arcspace has incompressible boundary, or a knot with an essential n-string tangle decomposition, where (II) each tangle has no parallel strings, or (III) one tangle space is not a handlebody and K is not cabled, then any nontrivial surgery on every component of K produces irreducible manifolds in all cases (with some exceptional surgeries in case (I)) and, in particular, Haken manifolds in cases (I) and (III). Moreover, if K is hyperbolic in (III) and at least one tangle space has incompressible boundary, then all nontrivial surgeries on K are also hyperbolic; this last result is also established for type (I) decompositions under some constraints.