# The $\nu^+$-equivalence classes of genus one knots

## TL;DR

This paper classifies genus one knots into three $
u^+$-equivalence classes, showing they are all equivalent to the trefoil, its mirror, or the unknot, based on invariants from Heegaard Floer theory.

## Contribution

It establishes a complete classification of genus one knots under $
u^+$-equivalence, linking them to well-known knots using knot Floer invariants.

## Key findings

- Genus one knots are $
u^+$-equivalent to trefoil, mirror, or unknot.
- $
u^+$-equivalence preserves many Heegaard Floer invariants.
- Classification simplifies understanding of genus one knot concordance classes.

## Abstract

The $\nu^+$-equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes $CFK^{\infty}$, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this paper, we show that any genus one knot is $\nu^+$-equivalent to one of the trefoil, its mirror and the unknot.