Primary decomposition and the fractal nature of knot concordance

Joint with Tim Cochran and Shelly Harvey

Mathematische Annalen, 351 (2011), 443-508

For each sequence P = (p_1(t), p_2(t),...) of polynomials 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 a sequence of higher-order Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the n-solvable filtration. 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.


2-torsion in the n-solvable filtration of the knot concordance group

Joint with Tim Cochran and Shelly Harvey

Proceedings of the London Mathematical Society, 102 (2011), 257-290

Cochran-Orr-Teichner introduce a natural filtration F_n of the smooth knot concordance group called the n-solvable filtration. We show that each associated graded abelian group G_n = F_n/F_{n.5} contains infinite linearly independent sets of elements of order 2 (this was known previously for n=0,1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, tau-invariant, delta-invariants and Casson-Gordon invariants. Moreover each is slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in G_n, one for each distinct n-tuple P = (p_1(t),...,p_n(t)) of knot polynomials. Such a sequence of polynomials records the orders of certain submodules of a sequence of higher-order Alexander modules of the knot.


Derivatives of knots and second-order signatures

Joint with Tim Cochran and Shelly Harvey

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, 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.


L^2-Betti numbers of plane algebraic curves

Joint with Stefan Friedl and Laurentiu Maxim

Michigan Mathematical Journal, 58 (2009), 411-421

In 2007, M. W. Davis, T. Januszkiewicz and I. J. Leary showed that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non-zero. We will prove an analogous statement for complements of any algebraic curve in C^2. Furthermore we also recast and extend previous results of Leidy and Maxim in terms of L^2-Betti numbers.


Knot concordance and higher-order Blanchfield duality

Joint with Tim Cochran and Shelly Harvey

Geometry & Topology, 13 (2009), 1419-1482

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.


Link concordance and generalized doubling operators

Joint with Tim Cochran and Shelly Harvey

Algebraic & Geometric Topology, 8 (2008), 1593-1646

We introduce a new 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. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any rho invariants associated to solvable representations into finite unitary groups.


Obstructions on fundamental groups of plane curve complements

Joint With Laurentiu Maxim

Real and Complex Singularities, Contemporary Mathematics, 459 (2008), 117-130

We survey various Alexander-type invariants of plane curve complements and, in relation to a question of Serre, we emphasize certain obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. These obstructions are then discussed on special classes of examples. In particular, we give new explicit computations of higher-order degrees of curves, which are invariants defined in a previous paper of the authors.


Higher-order Alexander invariants of plane algebraic curves

Joint with Laurentiu Maxim

International Mathematics Research Notices, article ID 12976 (2006), 23 pages

We define new higher-order Alexander modules and higher-order degrees which are invariants of the algebraic planar curve. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve. These invariants are in the spirit of those developed by T. Cochran and S. Harvey, which were used to study knots, 3-manifolds, and finitely presented groups. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.


Higher-order linking forms for knots

Commentarii Mathematici Helvetici, 81 (2006), 755-781

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.