Derivatives of knots and second-order signatures

Abstract

We define a set of “second-order” $L^{\left(2\right)}$–signature invariants for any algebraically slice knot. These obstruct a knot’s being a slice knot and generalize Casson–Gordon invariants, which we consider to be “first-order signatures”. As one application we prove: If $K$ is a genus one slice knot then, on any genus one Seifert surface $\Sigma$, there exists a homologically essential simple closed curve $J$ of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new relation, generalizing homology cobordism, called null-bordism.