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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)
Keywords:
band on a link
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