Knots with algebraic unknotting number one
In [Quest. Answers Gen. Topology 8, 283-292 (1990; Zbl 0704.57004)], \textit{H. Murakami} introduced the notion of an algebraic unknotting operation on Seifert matrices and defined the algebraic unknotting number \(u_ a (K)\) of a knot \(K\) by using the notion. This satisfies the inequality \(u_ a (K) \leq u(K)\), where \(u(K)\) is the unknotting number of \(K\). In the paper under review, the author gives a characterization of knots with \(u_ a (K) = 1\). Namely, he proves that a knot with Alexander polynomial \(\Delta_ K\) has \(u_ a(K) = 1\) if and only if the Alexander module is cyclic and has a generator \(g\) with \(\beta (g,g) = \pm 1/ \Delta_ K\), where \(\beta\) is the Blanchfield pairing. The only if part had been proved by Murakami. As an application, the author shows that the knot \(8_{10}\), which is thought to have unknotting number 2, has algebraic unknotting number 1. This shows that abelian methods cannot be used to show its unknotting number is actually 2.
algebraic unknotting operation on Seifert matrices
scientific article; zbMATH DE number 562513