Volume 10, issue 2 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 5, 2509–3131
Issue 4, 1883–2507
Issue 3, 1259–1881
Issue 2, 635–1258
Issue 1, 1–633

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Derivatives of knots and second-order signatures

Tim D Cochran, Shelly Harvey and Constance Leidy

Algebraic & Geometric Topology 10 (2010) 739–787

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.

knot concordance, slice knot, $n$–solvable, signature
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M10
Received: 29 December 2008
Revised: 17 January 2010
Accepted: 17 January 2010
Published: 22 March 2010
Tim D Cochran
Department of Mathematics MS-136
Rice University
PO 1892
Houston, Texas 77251-1892
Shelly Harvey
Department of Mathematics MS-136
Rice University
PO 1892, Houston, Texas 77251-1892
Constance Leidy
Department of Mathematics
Wesleyan University
Wesleyan Station
Middletown, CT 06459