Derivatives of Knots and Second-order Signatures

Details Derivatives of Knots and Second-order Signatures Derivatives of Knots and Second-order Signatures

2008-08-11

arxiv.org/abs/0808.1432v2

Alg. Geom. Topology 10 (2010), 739-787

Mathematics

Geometric Topology

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

Scientific paper

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, 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.

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