Higher dimensional simple knots and minimal Seifert surfaces
A Dedekind knot is a simple (2q-1)-knot, \(q\geq 3\), for which the annihilator ideal is generated by an irreducible polynomial \(\lambda\) (X), and the ring \({\mathbb{Z}}[X,X^{-1}]/(\lambda)\) is Dedekind. The author begins by considering any simple (2q-1)-knot, \(q\geq 3\), and classifying the minimal Seifert surfaces (up to isotopy) in terms of isometry classes of certain unimodular \((-1)^{q+1}\)-Hermitian forms. By extending the scalars, each of these forms yields the Blanchfield pairing of the knot. As an application of this result, one obtains a short proof of a special case of a theorem of Trotter, that if the leading coefficient of the Alexander polynomial is prime, then the knot has only one minimal Seifert surface. Then simple (2q-1)-knots, \(q\geq 3\), are considered which are Dedekind and either non-fibred or have indefinite Blanchfield pairing. A necessary and sufficient condition for two such pairings to be isometric is given, in terms of rank, signatures and determinants. As a corollary, a cancellation theorem for these knots is obtained, and a necessary and sufficient condition for two Seifert surfaces of a given knot to be isotopic is also proved. Necessary and sufficient conditions are given for a Dedekind knot to be (-1)-amphicheiral, and for a non-fibred or indefinite Dedekind knot to be invertible.
scientific article; zbMATH DE number 3842771