  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.
