Classification of Links Up to 0-Solvability

Abstract

The $n$-solvable filtration of the $m$-component smooth (string) link concordance group, $$\cdots \subset {\mathcal{F}}_{n+1}^m\subset {\mathcal{F}}_{n.5}^m\subset {\mathcal{F}}_n^m\cdots \subset {\mathcal{F}}_1^m\subset {\mathcal{F}}_{0.5}^m\subset {\mathcal{F}}_0^m\subset {\mathcal{F}}_{-0.5}^m\subset {\mathcal{C}}^m,$$ as defined by Cochran, Orr, and Teichner, is a tool for studying smooth knot and link concordance that yields important results in low-dimensional topology. The focus of this paper is to give a characterization of the set of 0-solvable links. We introduce a new equivalence relation on links called 0-solve equivalence and establish both an algebraic and a geometric classification of ${\mathbb{L}}_0^m$, the set of links up to 0-solve equivalence. We show that ${\mathbb{L}}_0^m$ has a group structure isomorphic to the quotient ${\mathcal{F}}_{-0.5}/{\mathcal{F}}_0$ of concordance classes of string links and classify this group, showing that $${\mathbb{L}}_0^m\cong {\mathcal{F}}_{-0.5}^m/{\mathcal{F}}_0^m\cong {\mathbb{Z}}_2^m\oplus {\mathbb{Z}}^{\left(\genfrac{}{}{0ex}{}{m}{3}\right)}\oplus {\mathbb{Z}}_2^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)}.$$ Finally, using results of Conant, Schneiderman, and Teichner, we show that 0-solvable links are precisely the links that bound class 2 gropes and support order 2 Whitney towers in the 4-ball.

arXiv comment34 pages

References in corpus (6)

• Knot Concordance and Higher-Order Blanchfield Duality
• Homology Cobordism Invariants and the Cochran-Orr-Teichner Filtration of the Link Concordance Group
• Knot concordance, Whitney towers and L^2 signatures
• Higher Order Intersections in Low-Dimensional Topology
• Homology and Derived Series of Groups II: Dwyer's Theorem
• The (n)-solvable filtration of the link concordance group and Milnor's invariants

Cited by in corpus (1)

• Stably slice disks of links