Derivatives of Knots and Second-order Signatures

TL;DR

This paper introduces second-order L^2-signature invariants for algebraically slice knots, extending Casson-Gordon invariants, and explores their implications for slice knots and knot derivatives.

Contribution

It defines second-order signatures, introduces the concept of derivatives of knots with respect to metabolizers, and proposes a new null-bordism equivalence relation.

Findings

Second-order signatures obstruct sliceness of knots.

For genus one slice knots, certain signatures vanish on specific curves.

Introduces a new geometric notion and equivalence relation for knots.

Abstract

We define a set of "second-order" L^(2)-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, there exists a homologically essential simple closed curve 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 equivalence relation, generalizing homology cobordism, called null-bordism.