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Twisting and knot types. (English) Zbl 0739.57003

Let \(K\) be an unoriented smooth knot in the oriented 3-sphere \(S^ 3\), and \(V(\subset S^ 3)\) a solid torus with preferred framing which contains \(K\) in its interior. Let \(f_ n\) be an orientation preserving homeomorphism of \(V\) satisfying \(f_ n(m)=m\) and \(f_ n(\ell)=m+n\ell\) in \(H_ 1(\partial V)\), where \((m,\ell)\) is a meridian-longitude pair of \(V\). Then we can obtain a new knot \(f_ n(K)\) in \(S^ 3\), and we call this operation twisting. A twisting problem — Can twisting change knot type of \(K\) if the wrapping number of \(K\) in \(V\) is greater than or equal to two? — is discussed, and several answers are given.
Reviewer: K.Motegi

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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