A Higher-order Genus Invariant and Knot Floer Homology

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of \(K\) detects more structure of minimal genus Seifert surfaces for \(K\). We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian \(L^2\)-signatures bound this invariant from below.