INVARIANTS OF KNOTS DERIVED FROM EQUIVARIANT LINKING MATRICES OF THEIR SURGERY PRESENTATIONS

Abstract

The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.