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2010 Derivatives of knots and second-order signatures
Tim D Cochran, Shelly Harvey, Constance Leidy
Algebr. Geom. Topol. 10(2): 739-787 (2010). DOI: 10.2140/agt.2010.10.739


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.


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Tim D Cochran. Shelly Harvey. Constance Leidy. "Derivatives of knots and second-order signatures." Algebr. Geom. Topol. 10 (2) 739 - 787, 2010.


Received: 29 December 2008; Revised: 17 January 2010; Accepted: 17 January 2010; Published: 2010
First available in Project Euclid: 19 December 2017

zbMATH: 1192.57005
MathSciNet: MR2606799
Digital Object Identifier: 10.2140/agt.2010.10.739

Primary: 57M25
Secondary: 57M10

Keywords: $n$–solvable , knot concordance , signature , slice knot

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.10 • No. 2 • 2010
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