Tim D. Cochran

Noncommutative knot theory

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S 3 - K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure of G ( 1 ) /G ( 2 ) as a module over G/G ( 1 ) (here G ( n ) is the n t h term of the derived series of G). Hence any phenomenon associated to G ( 2 ) is invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the n t h higher-order Alexander module, G ( + 1 ) /G ( n + 2 ) , considered as a Z[G/G ( n + 1 ) ]-module. We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4-manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3-manifolds.

Geometric invariants of link cobordism

AbstractA geometric notion of a “derivative” is defined for 2-component links ofSn inSn+2 and used to construct a sequenceβi,i=1,2,... of abelian concordance invariants which vanish for boundary links. Forn>1, these generalize the only heretofore known invariant, the Sato-Levine invariant. Forn=1, these invariants are additive under any band-sum and consequently provide new information about which 1-links are concordant to boundary links. Examples are given of concordance classes successfully distinguished by theβi but not by their

\rho

\bar \mu

Knot concordance and higher-order Blanchfield duality

, Murasugi 2-height, Sato-Levine invariant or Alexander polynomial.

Primary decomposition and the fractal nature of knot concordance

We present new results, announced in [T], on the classical knot concordance group C. We establish the nontriviality at all levels of the (n)-solvable filtration · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0 ⊆ C introduced in [COT1]. Recall that this filtration is significant due to its intimate connection to tower constructions arising in work of A. Casson and M. Freedman on the topological classification problem for 4-manifolds and due to the fact that all previously known concordance invariants are reflected in the first few terms in the filtration. In [COT1], nontriviality at the first new level n = 3 was established. Here, we prove the nontriviality of the filtration for all n, hence giving the ultimate justification to the theory. A broad range of techniques is employed in our proof, including cut-and-paste topology and analytical estimates. We use the Cheeger-Gromov estimate for von Neumann ρ-invariants, a deep analytic result. We also introduce a number of new algebraic arguments involving noncommutative localization and Blanchfield forms. We have attempted to make this article accessible to readers with only passing knowledge of [COT1].

Link concordance invariants and homotopy theory

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

Finite type invariants of 3-manifolds

For each sequence