Conjugacy search problem and the Andrews-Curtis conjecture
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.
Cite
@article{arxiv.1609.00325,
title = {Conjugacy search problem and the Andrews-Curtis conjecture},
author = {Dmitry Panteleev and Alexander Ushakov},
journal= {arXiv preprint arXiv:1609.00325},
year = {2016}
}