# A full-twist inequality for the $\nu^+$-invariant

## TL;DR

This paper establishes a comprehensive full-twist inequality for the $
u^+$-invariant, extending Wu's cabling formula to all cables and exploring the structure of $
u^+$-equivalence classes within knot concordance.

## Contribution

It introduces a full-twist inequality for $
u^+$, generalizes Wu's cabling formula, and studies the partial order on $
u^+$-equivalence classes.

## Key findings

- Full-twist inequality for $
u^+$ established
- Extended Wu's cabling formula to all cables
- Analyzed the partial order on $
u^+$-equivalence classes

## Abstract

Hom and Wu introduced a knot concordance invariant called $\nu^+$, which dominates many concordance invariants derived from Heegaard Floer homology. In this paper, we give a full-twist inequality for $\nu^+$. By using the inequality, we extend Wu's cabling formula for $\nu^+$ (which is proved only for particular positive cables) to all cables in the form of an inequality. In addition, we also discuss $\nu^+$-equivalence, which is an equivalence relation on the knot concordance group. We introduce a partial order on $\nu^+$-equivalence classes, and study its relationship to full-twists.