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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