Every genus one algebraically slice knot is 1-solvable

Abstract

Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by ${\mathcal{F}}_n$. It has been shown that ${\mathcal{F}}_n/{\mathcal{F}}_{n.5}$ is a very large group for $n\ge 0$. For a generalization to the setting of links the third author showed that ${\mathcal{F}}_{n.5}/{\mathcal{F}}_{n+1}$ is non-trivial. In this paper we provide evidence that for knots ${\mathcal{F}}_{0.5}={\mathcal{F}}_1$. In particular we prove that every genus 1 algebraically slice knot is 1-solvable.

19 pages, 10 figures, to appear in Transactions of the American Mathematical Society