Conjugacy search problem and the Andrews-Curtis conjecture

TL;DR

This paper introduces new computational methods and algorithms to analyze the Andrews-Curtis conjecture, focusing on potential counterexamples and enhancing search strategies within the space of balanced group presentations.

Contribution

It develops a new transformation called ACM-move, studies the search space modulo automorphisms, and applies these methods to AK(n) presentations, advancing the computational approach to the conjecture.

Findings

Unable to trivialize AK(n) for n>2 despite extensive efforts.

Introduced ACM-move to improve search space properties.

Proved automorphism-moves can be applied to AK(n) presentations.

Abstract

We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK(3). To improve metric properties of the search space (which is a set of balanced presentations of the trivial group) we introduce a new transformation (called an ACM-move here) that generalizes the original Andrews-Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. Finally, we prove that automorphism-moves can be applied to AK(n)-presentations. Unfortunately, despite a lot of effort we were unable to trivialize any of AK(n)-presentations, for n>2.