The unknotting number and classical invariants I

TL;DR

This paper introduces a new invariant derived from the Blanchfield pairing that provides a stronger lower bound on the unknotting number of knots, surpassing previous invariants and enabling analysis of complex knots.

Contribution

The authors develop a novel invariant from the Blanchfield pairing that improves lower bounds on unknotting numbers and applies to knots previously difficult to analyze.

Findings

New invariant from the Blanchfield pairing established.

Lower bounds on unknotting number improved over existing invariants.

Determined unknotting number at least three for 25 complex knots.

Abstract

Given a knot K we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of K. This lower bound subsumes the lower bounds given by the Levine-Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for 25 knots with up to 12 crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise.