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Tangles, property P, and a problem of J. Martin. (English) Zbl 0563.57002
A band on a link $$L\subset S^ 3$$ is an embedding $$b: I\times I\to S^ 3$$ such that $$b^{-1}(L)=\partial I\times I$$. A band b is trivial if there is a disk in $$S^ 3$$ whose interior is disjoint from $$L\cup image(b)$$ and whose boundary is the union of b(I$$\times \{0\})$$ and a subarc of L. Say $$L_ n$$ is obtained by an n-twist banding of L if $$L_ n$$ is obtained by replacing the two arcs b($$\partial I\times I)$$ of a band b on L with a pair of arcs near b.
Theorem: If K and $$K_ n$$ are both the unknot, and $$| n| \geq 2$$, then the band is trivial. In the notation of the Conway calculus we have, using $$n=2:$$ Corollary 1: If $$L_+$$ and $$L_-$$ are the unknot, $$L_ 0$$ is the unlink. Corollary 2: An amphicheiral, strongly invertible knot has property P. Corollary 1 solves problem 1.18 on the 1978 Kirby problem list. The case $$| n| =1$$ is far more difficult, but true, and is the subject of a later paper. Its solution removes ”amphicheiral” from Corollary 2.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)