Inequivalent fibred knots whose homotopy Seifert pairings are isometric

The problem of classification in terms of a finite list of invariants is a fundamental question of knot theory. In 1970, Levine I-L] gave an algebraic classification of n-dimensional knots in S n+2 bounding r-connected Seifert surfaces, where 2 r+ 1 = n, n> 3. Levine showed that the only invariant which determines the isotopy type of such knots is the homology Seifert pairing considered up to S-equivalence. In the mid 1970's, Kearton [K] and Trot ter [Tr 1; Tr 2] obtained the same classification in terms of the Blanchfield pairing. In 1980, Farber I-F] extended this classification in homotopy theoretic terms to n-knots that bound r-connected Seifert surfaces, where 3r > n + 1. Suppose V" § 1 ~ S" + 2 is a Seifert surface for an n-knot. Farber introduces a map 6): V^ V ~ S "§ 1, called the homotopy Seifert pairing of V, which induces the usual Seifert pairing on homology. He shows that the isometry class of ~ determines the isotopy class of V as a submanifold of S" § 2 in a certain range of dimensions and connectivities: