Higher-order linking forms for knots

We construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms, showing that the linking forms provide more information than the modules alone. This generalizes work of Trotter, who found examples of knots that have isomorphic classical Alexander modules, but non-isomorphic classical Blanchfield linking forms.Comment: 22 pages, 2 figure

Twisted Blanchfield pairings and decompositions of 3-manifolds

We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-manifold Y with a representation ϕ:Z[π1(Y)]→R, infected by a knot J along a curve η with ϕ(η)≠1, splits orthogonally as the sum of the twisted Blanchfield pairing of Y and the ordinary Blanchfield pairing of the knot J, with the latter tensored up from Z[t,t−1] to R

Derivatives of Knots and Second-order Signatures

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.Comment: 40 pages, 22 figures, typographical corrections, to appear in Alg. Geom. Topolog

Knot Concordance and Higher-Order Blanchfield Duality

In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} of the classical knot concordance group C. The filtration is important because of its strong connection to the classification of topological 4-manifolds. Here we introduce new techniques for studying C and use them to prove that, for each natural number n, the abelian group F_n/F_{n.5} has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson-Gordon and Gilmer, contain slice knots.Comment: Corrected Figure in Example 8.4, Added Remark 5.11 pointing out an important strengthening of Theorem 5.9 that is needed in a subsequent pape

Link concordance and generalized doubling operators

We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. We define generalized doubling operators, of which Bing doubling is an instance, and prove our nontriviality results in this more general context. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group.Comment: 45 pages. Final version. Changed figures 1.3 and 4.2. Expanded Remark 5.4. Fixed typos and made other minor changes. Some of the results are renumbered. Updates references. Note: All results except Cor. 4.8, Ex. 4.4, Ex. 4.6, Lemmas 6.4, 6.5 appeared previously in 0705.3987 under different title: Knot concordance and Blanchfield dualit

Primary decomposition and the fractal nature of knot concordance

For each sequence of polynomials, P=(p_1(t),p_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S^3, such a sequence of polynomials arises naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of Cochran-Orr-Teichner. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no Cochran-Orr-Teichner knot is concordant to any Cochran-Harvey-Leidy knot.Comment: 60 pages, added 4 pages to introduction, minor corrections otherwise; Math. Annalen 201

Higher-order linking forms

Trotter [T] found examples of knots that have isomorphic classical Alexander modules, but non-isomorphic classical Blanchfield linking forms. T. Cochran [C] defined higher-order Alexander modules, An , (K), of a knot, K, and higher-order linking forms, Bℓn (K), which are linking forms defined on An , (K). When n = 0, these invariants are just the classical Alexander module and Blanchfield linking form. The question was posed in [C] whether Trotter's result generalized to the higher-order invariants. We show that it does. That is, we construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms. Furthermore, we define new higher-order linking forms on the Alexander modules for 3-manifolds considered by S. Harvey [H]. We construct examples of 3-manifolds with isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms