The unknotting number and classical invariants II

TL;DR

This paper relates a knot invariant derived from the Blanchfield form to Levine-Tristram signatures and nullities, providing new insights into the unknotting number and showing the diagonalizability of the Blanchfield form over the reals.

Contribution

It expresses the invariant n_R(K) in terms of classical signatures and nullities, and proves the diagonalizability of the Blanchfield form over the real numbers.

Findings

n_R(K) can be expressed via Levine-Tristram signatures and nullities

Blanchfield form over reals is diagonalizable

Provides lower bounds on the unknotting number

Abstract

In [BF12] the authors associated to a knot K an invariant n_R(K) which is defined using the Blanchfield form and which gives a lower bound on the unknotting number. In this paper we express n_R(K) in terms of Levine-Tristram signatures and nullities of K. In the proof we also show that the Blanchfield form with real coefficients is diagonalizable.