Georgia Institute of Technology College of Sciences

Every knot in S^3 bounds a PL (piecewise-linear) disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus? I will discuss the joint work with Hom and Stoffregen, where we proved that for each n, there exists a knot K_n in an integer homology sphere that does not bound a PL surface of genus n in any homology four ball. This talk is meant to be accessible to a broad audience.