The Relation Between Diagrams of a Knot and Its Unknotting Number

TL;DR

This paper explores the relationship between knot diagrams and the unknotting number, aiming to determine if the number can be derived from a single diagram, but concludes that Reidemeister moves can alter this number, complicating such determination.

Contribution

The paper investigates whether the unknotting number can be obtained from any single diagram and examines the invariance under Reidemeister moves, ultimately showing the difficulty in such an approach.

Findings

Reidemeister II move can change the unknotting number

The attempt to prove invariance under Reidemeister moves was unsuccessful

The paper is withdrawn due to the invalidity of the proposition

Abstract

The unknotting number is the classical invariant of a knot. However, its determination is difficult in general. To obtain the unknotting number from definition one has to investigate all possible diagrams of the knot. We tried to show the unknotting number can be obtained from any one diagram of the knot. To do this we tried to prove the unknotting number is not changed under Riedemiester moves, but such a proposition is not correct. Reidemeister II move can change unknotting number. See Nakanishi-Bleiler example. So this article is withdrawn.