Breadth-First Search and the Andrews–curtis Conjecture

Havas, George; Ramsay, Colin M.

doi:10.1142/s0218196703001365

Cited by 23 publications

(17 citation statements)

References 10 publications

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“…Hence, the inequality (3.1) could not be satisfied. On the other hand, as was found out by Myasnikov [17], see also [5], [18], the AC-conjecture holds for the pair (a 2 b −3 , abab −1 a −1 b −1 ). Therefore, the pair (a 2 b −3 , abab −1 a −1 b −1 ) is not minimal and gives a counterexample to [2,Conjecture 4].…”

Section: One More Conjecture Of Andrews and Curtissupporting

confidence: 58%

On Conjectures of Andrews and Curtis

Ivanov

2018

Proc. Amer. Math. Soc.

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It is shown that the original Andrews-Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis [2] made in 1966. We also consider a more restrictive "cancellative" version of the cyclic Andrews-Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews-Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.

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Section: One More Conjecture Of Andrews and Curtissupporting

confidence: 58%

On Conjectures of Andrews and Curtis

Ivanov

2018

Proc. Amer. Math. Soc.

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“…Recently, Havas and Ramsay [11] showed that this is, in fact, the only (up to AC-equivalence) possible counterexample of length 13. We were not able to crack this example, but the following fact might be of interest:…”

Section: Introductionmentioning

confidence: 96%

On the Andrews-Curtis equivalence

Myasnikov¹,

Shpilrain²

2002

Contemporary Mathematics

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Abstract. The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of "elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the previously proposed examples are actually not counterexamples. We hope that the tricks we used in constructing relevant chains of elementary transformations will be useful to those who attempt to establish the Andrews-Curtis equivalence in other situations.On the other hand, we give two rather general and simple methods for constructing balanced presentations of the trivial group; some of these presentations can be considered potential counterexamples to the Andrews-Curtis conjecture. One of the methods is based on a simple combinatorial idea of composition of group presentations, whereas the other one uses "exotic" knot diagrams of the unknot.We also consider the Andrews-Curtis equivalence in metabelian groups and reveal some interesting connections of relevant problems to well-known problems in K-theory.

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“…We also use Tietze transformation-based approaches. We have a standalone program automac (automorphic Andrews-Curtis) which is based on ACME (Andrews-Curtis Move Enumerator) [16]. It combines length-preserving automorphisms of the extended symmetric group [15] and some Whitehead automorphisms with Andrews-Curtis moves.…”

Section: Detailsmentioning

confidence: 99%