Knot genus is a fundamental quantity in knot theory. While the 3–genus of a knot is theoretically computable, 4–genus is more elusive. The situation is even more complicated for knots in 3–manifolds other than the 3–sphere. Yet, knot Floer homology contains many insights. I'll discuss genus bounds coming from a particular Floer theoretic invariant called "tau." These bounds have applications in different directions including link concordance, rational concordance and contact topology.
This is joint work with Matthew Hedden.