Higher-order Alexander invariants and filtrations of the knot concordance group

Abstract:

We establish certain “nontriviality” results for several filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group, $\mathcal {C}$, defined by K. Orr, P. Teichner and the first author: \[ 0\subset \cdots \subset \mathcal {F}_{(n.5)}\subset \mathcal {F}_{(n)}\subset \cdots \subset \mathcal {F}_{(1.5)}\subset \mathcal {F}_{(1.0)}\subset \mathcal {F}_{(0.5)} \subset \mathcal {F}_{(0)}\subset \mathcal {C},\] we refine the recent nontriviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related symmetric Grope filtration of $\mathcal {C}$. We also show that the Grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the Ozsváth-Szabó $\tau$-invariant nor by J. Rasmussen’s $s$-invariant. Our broader contribution is to establish, in “the relative case”, the key homological results whose analogues Cochran-Orr-Teichner established in “the absolute case”. We say two knots $K_0$ and $K_1$ are concordant modulo $n$-solvability if $K_0\#(-K_1)\in \mathcal {F}_{(n)}$. Our main result is that, for any knot $K$ whose classical Alexander polynomial has degree greater than 2, and for any positive integer $n$, there exist infinitely many knots $K_i$ that are concordant to $K$ modulo $n$-solvability, but are all distinct modulo $n.5$-solvability. Moreover, the $K_i$ and $K$ share the same classical Seifert matrix and Alexander module as well as sharing the same higher-order Alexander modules and Seifert presentations up to order $n-1$.