The Andrews-Curtis conjecture, proposed in the 1950s, asserts that all balanced presentations of the trivial group can be simplified through specific transformations known as stable Andrews-Curtis transformations to the trivial presentation. This conjecture is connected to the smooth 4-dimensional Poincaré conjecture, and some believe that both conjectures may be incorrect. In my talk, however, I will explain a partial connection between the Andrews-Curtis equivalence and a notion of equivalence on certain families of 3-manifolds, and I will demonstrate that a version of the conjecture is not true.