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Conjugacy search problem and the Andrews-Curtis conjecture

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arxiv 1609.00325 v1 pith:BZQORHH5 submitted 2016-09-01 math.GR

Dmitry Panteleev , Alexander Ushakov This is my paper
classification math.GR
keywords presentationsandrews-curtissearchspacebalancedconjecturegroupintroduce
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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  1. Machine-checkable equivalence certificates at the length-14 Andrews-Curtis frontier

    math.GR 2026-07 accept novelty 7.0 of 10

    Explicit elementary-move certificates prove two MS(3) presentations AC-equivalent to AK(3) and realize the automorphism σ on the two remaining open MS(2) classes at length 14.

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