On the algebraic unknotting number

TL;DR

This paper proves that the algebraic unknotting number of a knot equals a previously defined invariant based on the Blanchfield form, unifying all classical lower bounds into a single exact measure.

Contribution

The authors establish that the invariant n(K) derived from the Blanchfield form exactly equals the algebraic unknotting number u_a(K), confirming its status as a precise measure.

Findings

n(K) equals u_a(K) for all knots

n(K) subsumes all classical lower bounds

Provides a complete algebraic characterization of unknotting number

Abstract

The algebraic unknotting number u_a(K) of a knot K was introduced by Hitoshi Murakami. It equals the minimal number of crossing changes needed to turn K into an Alexander polynomial one knot. In a previous paper the authors used the Blanchfield form of a knot K to define an invariant n(K) and proved that n(K) is a lower bound on u_a(K). They also showed that n(K) subsumes all previous classical lower bounds on the (algebraic) unknotting number. In this paper we prove that n(K)=u_a(K).