Research
Primary decomposition and the fractal nature of knot concordanceJoint with Tim Cochran and Shelly Harvey Mathematische Annalen, 351 (2011), 443508 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 higherorder Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the nsolvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higherorder 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 CochranOrrTeichner knot is concordant to any CochranHarveyLeidy knot. 

2torsion in the nsolvable filtration of the knot concordance groupJoint with Tim Cochran and Shelly Harvey Proceedings of the London Mathematical Society, 102 (2011), 257290 CochranOrrTeichner introduce a natural filtration F_n of the smooth knot concordance group called the nsolvable 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 sinvariant, tauinvariant, deltainvariants and CassonGordon invariants. Moreover each is slice in a rational homology 4ball. In fact we show that there are many distinct such classes in G_n, one for each distinct ntuple 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 higherorder Alexander modules of the knot. 

Derivatives of knots and secondorder signaturesJoint with Tim Cochran and Shelly Harvey Algebraic & Geometric Topology, 10 (2010), 739787 We define a set of secondorder L^2signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize CassonGordon invariants, which we consider to be firstorder 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 selflinking zero, which has vanishing zeroth order signature and a vanishing firstorder 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 nullbordism. 

L^2Betti numbers of plane algebraic curvesJoint with Stefan Friedl and Laurentiu Maxim Michigan Mathematical Journal, 58 (2009), 411421 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^2Betti numbers of its complement is nonzero. 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^2Betti numbers. 

Knot concordance and higherorder Blanchfield dualityJoint with Tim Cochran and Shelly Harvey Geometry & Topology, 13 (2009), 14191482 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 4manifolds. 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 longstanding question as to whether certain natural families of knots, first considered by CassonGordon and Gilmer, contain slice knots. 

Link concordance and generalized doubling operatorsJoint with Tim Cochran and Shelly Harvey Algebraic & Geometric Topology, 8 (2008), 15931646 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 CheegerGromov bound, a deep analytical tool used by CochranTeichner. 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 complementsJoint With Laurentiu Maxim Real and Complex Singularities, Contemporary Mathematics, 459 (2008), 117130 We survey various Alexandertype 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 higherorder degrees of curves, which are invariants defined in a previous paper of the authors. 

Higherorder Alexander invariants of plane algebraic curvesJoint with Laurentiu Maxim International Mathematics Research Notices, article ID 12976 (2006), 23 pages We define new higherorder Alexander modules and higherorder 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, 3manifolds, and finitely presented groups. We show that for curves in general position at infinity, the higherorder 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. 

Higherorder linking forms for knotsCommentarii Mathematici Helvetici, 81 (2006), 755781 We construct examples of knots that have isomorphic nthorder Alexander modules, but nonisomorphic nthorder 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 nonisomorphic classical Blanchfield linking forms. 