Twisted unknots.
Let \(D\) be a disk in the three-sphere and \(K\) a knot that meets \(D\) transversally in its interior. Let \(n\) be an integer and \(K_{D,n}\) the knot obtained from \(K\) by \(n\)-twisting along the disk \(D\). If \(K\) is trivial, then \((K,D)\) is called a twisting pair and the knot \(K_{D,n}\) is called a twisted unknot. It is well known from the Thurston uniformization theorem and the torus theorem that every knot in the three-sphere is a torus knot, a satellite knot or a hyperbolic knot. The paper under review discusses the geometric conditions under which a twisted unknot is a torus, satellite or hyperbolic knot. Namely, it is proved that: If \((K,D)\) is a hyperbolic pair, then \(K_{D,n}\) is a hyperbolic knot for any integer \(n\), such that \(| n| >1\). If \((K,D)\) is a Seifert fibered pair, then \(K_{D,n}\) is a torus knot for any integer \(n\). The paper considers also the case where \((K,D)\) is a toroidal pair, and gives a description of the pairs \((K,n)\) for which \(K_{D,n}\) is not a satellite knot.
scientific article; zbMATH DE number 1995409
scientific article