Georgia Institute of Technology College of Sciences

We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.