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 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.57M25; 57M10

Smooth slice boundary links whose derivative links have nonvanishing Milnor invariants

“…The following archetypal theorem of [CHL10] is one of the strongest known; it follows previous similar theorems of D. Cooper (from his thesis but unpublished), Gilmer [Gil83,Gil93] and Cochran-Orr-Teichner [COT04]. Theorem 1.3 (Cochran-Harvey-Leidy [CHL10]).…”

“…The following archetypal theorem of [CHL10] is one of the strongest known; it follows previous similar theorems of D. Cooper (from his thesis but unpublished), Gilmer [Gil83,Gil93] and Cochran-Orr-Teichner [COT04]. Theorem 1.3 (Cochran-Harvey-Leidy [CHL10]). 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 that has vanishing zero-th order signature and a vanishing first order signature.…”

“…Following [CHL10], we consider simple closed curves on the Seifert surfaces representing a metaboliser M as a link in S 3 , also denoted by M , and call this the derivative of L with respect to the metaboliser.…”

The average signature of graph links

“…In general, ρ-invariants obstruct sliceness and give insight into the structure of the topological concordance group. We refer to [9,6,10] for some exemplary applications; in fact, the literature on the subject is now very vast. We also point out that the average signature of links with pairwise linking number 0 was studied to detect sliceness of some knots, see [6,10,11].…”

Von Neumann rho invariants and torsion in the topological knot concordance group

“…It is shown in [18] that this quotient is isomorphic to Z 1˚Z1 2˚Z 1 4 . In particular, this shows that the concordance group has infinite rank.…”