## Algebraic & Geometric Topology

### Derivatives of knots and second-order signatures

#### 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 $Σ$, 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 739-787.

Dates
Revised: 17 January 2010
Accepted: 17 January 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715113

Digital Object Identifier
doi:10.2140/agt.2010.10.739

Mathematical Reviews number (MathSciNet)
MR2606799

Zentralblatt MATH identifier
1192.57005

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M10: Covering spaces

#### Citation

Cochran, Tim D; Harvey, Shelly; Leidy, Constance. Derivatives of knots and second-order signatures. Algebr. Geom. Topol. 10 (2010), no. 2, 739--787. doi:10.2140/agt.2010.10.739. https://projecteuclid.org/euclid.agt/1513715113

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