#### Volume 10, issue 2 (2010)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Derivatives of knots and second-order signatures

### Tim D Cochran, Shelly Harvey and Constance Leidy

Algebraic & Geometric Topology 10 (2010) 739–787
##### 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.

##### Keywords
knot concordance, slice knot, $n$–solvable, signature
Primary: 57M25
Secondary: 57M10