On Fox's congruence classes of knots. II
[For Part I by the author and \textit{S. Suzuki} see ibid. 24, 217-225 (1985; Zbl 0654.57003).] In ``Congruence classes of knots'' [Osaka Math. J. 10, 37-41 (1958; Zbl 0084.192)], \textit{R. H. Fox} defined (n,q) congruence of knots by means of what is now called P/Q surgery, where \(P=1\) and certain conditions on the surgery are satisfied. Congruence generalizes to links and generates an equivalence relation among link types. Some theorems the author proves are: Theorem 2. Let \(n>0\), \(q>0\), (n,q)\(\neq (2,1),(2,2)\). For every such (n,q) and every \(\mu\geq 1\) there exist infinitely many distinct classes of \(\mu\)-component links which are inequivalent mod(n,q). Theorem 3. Two links with the same number of components are congruent mod(2,1) if and only if the components may be ordered so that each pair of corresponding components of each link have the same linking numbers mod 2. Corollary. All knot types are congruent mod(2,1). The author shows the Borromean rings and a 3-component trivial link are not congruent mod(2,2).
scientific article; zbMATH DE number 4157493