A null-homologous knot in a 3-manifold bounds a surface in the 3-manifold and, if you can compute the knot Floer homology, you can compute the minimal genus of such a surface. However, if we realize the 3-manifold as the boundary of some 4-manifold, the knot may bound a surface of smaller genus in the 4-manifold. While we can’t always compute this minimal genus precisely, we can use tools from Heegaard Floer theory, particularly the tau-invariant, to produce lower bounds. I will discuss the tau-invariant, a recent generalization due to Hedden and myself, and corresponding genus bounds. I will also discuss some applications to links and contact manifolds.
This is joint work with Matthew Hedden.