Every genus one algebraically slice knot is 1-solvable

TL;DR

This paper proves that all genus 1 algebraically slice knots are 1-solvable, providing evidence that the initial levels of the solvable filtration coincide for knots, which advances understanding of knot concordance.

Contribution

It establishes that every genus 1 algebraically slice knot is 1-solvable, suggesting the equality of certain filtration levels in knot concordance.

Findings

All genus 1 algebraically slice knots are 1-solvable.

Supports the conjecture that .5 and 1 levels of the filtration coincide for knots.

Advances understanding of the structure of the knot concordance group.

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.