Simple spun knots
The spin of a simple (2q-1)-knot is a simple 2q-knot. Using known classification results for these knots, this paper gives a complete description of the spinning operation for \(q\geq 5\). A simple (2q-1)-knot k is classified by a \({\mathbb{Z}}[t,t^{-1}]\)-module A(k) with a certain non-singular \(\pm\)-Hermitian form on A. A simple 2q-knot is classified by an ''F-form'' which includes a \({\mathbb{Z}}[t,t^{-1}]\)-module B(k). If \(\sigma\) (k) is the spin of k, a simple (2q-1)-knot, then the F-form of \(\sigma\) (k) is hyperbolic and \(B(\sigma\) (k)) is isomorphic to A(k). Conversely, if a simple 2q-knot \(\ell\) has hyperbolic F-form and \(B(\ell)\) is isomorphic to A(k), for some simple (2q-1)-knot k, then \(\ell =\sigma(k)\). One interesting consequence is that \(\sigma(k_ 1)=\sigma(k_ 2)\) if and only if the complement of \(k_ 1\) is homotopy equivalent to the complement of \(k_ 2\).
scientific article; zbMATH DE number 3848240