Unknotting Unknots

TL;DR

This paper explores the problem of unknotting unknots using arc-presentations, providing bounds on the complexity and number of moves needed to simplify knot diagrams, based on Dynnikov's work.

Contribution

It introduces quadratic bounds on crossing increases and move counts for unknotting sequences using Dynnikov's arc-presentation approach.

Findings

Quadratic upper bound on crossings introduced during unknotting

Upper bound on the number of moves needed for unknotting

Application of Dynnikov's arc-presentation to knot simplification

Abstract

A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surprisingly difficult, since it has been shown that knot diagrams may need to be made more complicated before they may be simplified. We do not yet know, however, how much more complicated they must get. We give an introduction to the work of Dynnikov who discovered the key use of arc--presentations to solve the problem of finding a way to detect the unknot directly from a diagram of the knot. Using Dynnikov's work, we show how to obtain a quadratic upper bound for the number of crossings that must be introduced into a sequence of unknotting moves. We also apply Dynnikov's results to find an upper bound for the number of moves required in an unknotting sequence.