Constance Leidy

Knot concordance and higher-order Blanchfield duality

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 n2 N0 , the group Fn=Fn: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. 57M25; 57M10

Primary decomposition and the fractal nature of knot concordance

For each sequence

Higher-order linking forms for knots

{\mathcal{P}=(p_1(t),p_2(t),\dots)}

Higher-order Alexander invariants of plane algebraic curves

of polynomials we define a characteristic series of groups, called the derived series localized at

L 2-Betti numbers of plane algebraic curves

{\mathcal{P}}

Obstructions on Fundamental Groups of Plane Curve Complements

. 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. The new filtrations allow us to distinguish between knots whose classical Alexander polynomials are coprime and even to distinguish between knots with coprime higher-order Alexander polynomials. This provides evidence of higher-order analogues of the classical 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.