On the Minkowski units of 2-periodic knots (Q2756092)

A knot \(K\) in the standard 3-sphere \(S^3\) is said to be \(n\)-periodic, \(n\geq 2\), if there exists a \(\mathbb{Z}_n\)-action on the pair \((S^3,K)\) such that the fixed point set \(F\) of the action is a circle in \(S^3\) disjoint from \(K\). It is well known that \(F\) is unknotted. So the quotient map \(p:S^3\to S^3/ \mathbb{Z}_n\) is an \(n\)-fold cyclic branched covering branched over \(p(F)=F'\), and \(p(K)=K'\) is also a knot in the orbit space \(S^3/ \mathbb{Z}_n\cong S^3\). The knot \(K'\) is called the factor knot of \(K\), and the 2-component link \(L=K'\cup F'\) is said to be associated to the \(n\)-periodic knot \(K\). The author gives a relationship among the Minkowski units, for all odd prime number including \(\infty\), of a 2-periodic knot \(K\) in the 3-sphere, its factor knot \(K'\) and the associated 2-component link \(L=K'\cup F'\). We recall that the Minkowski unit for a tame knot was first defined by \textit{L. Goeritz} for odd prime integers in [Math. Z. 36, 647-654 (1933; Zbl 0006.42201)], and then it was extended by \textit{K. Murasugi} for any oriented tame link and for any prime integer (including \(\infty)\) in [Trans. Am. Math. Soc. 117, 387-422 (1965; Zbl 0137.17903)].