Some comparisons of Blanchfield pairings and cohomology pairings of knots

TL;DR

This paper compares two types of pairings associated with knots, demonstrating their invariance properties and identifying when they are equivalent or not, thus contributing to knot invariants understanding.

Contribution

It provides a criterion for relating cohomology and Blanchfield pairings and analyzes their equivalence for different knots.

Findings

The cohomology pairing is an S-equivalent invariant.

Some knots have equivalent pairings, others do not.

Criteria are established for pairing relations.

Abstract

We study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an $S$-equivalent invariant, and give a criterion to a relation between the two pairings. We also observe that the pairings of some knots are equivalent, and that the pairings of other knots are not equivalent.