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, there exists a homologically essential simple closed curve 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 equivalence relation, generalizing homology cobordism, called null-bordism.

40 pages, 22 figures, typographical corrections, to appear in Alg. Geom. Topology

References in corpus (3)

• Knot Concordance and Higher-Order Blanchfield Duality
• Homology and Derived p-Series of Groups
• Link concordance and generalized doubling operators

Cited by in corpus (5)

• Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance
• A second order algebraic knot concordance group
• Von Neumann rho invariants and torsion in the topological knot concordance group
• Slice knots which bound punctured Klein bottles
• Action of the Mazur pattern up to topological concordance