# Knot traces and concordance

## TL;DR

This paper introduces a method to construct knot pairs with diffeomorphic 4-manifolds and uses Heegaard Floer invariants to distinguish their concordance classes, disproving a previous conjecture and exploring knot pattern actions.

## Contribution

It provides a novel construction of knot pairs with diffeomorphic 4-manifolds and demonstrates the use of d-invariants to obstruct smooth concordance, challenging existing conjectures.

## Key findings

- Constructed many knot pairs with diffeomorphic 4-manifolds.
- Used d-invariants to obstruct smooth concordance.
- Proved existence of invertible patterns with non-concordant images.

## Abstract

We give a method for constructing many pairs of distinct knots $K_0$ and $K_1$ such that the two 4-manifolds obtained by attaching a 2-handle to $B^4$ along $K_i$ with framing zero are diffeomorphic. We use the d-invariants of Heegaard Floer homology to obstruct the smooth concordance of some of these $K_0$ and $K_1$, thereby disproving a conjecture of Abe in [Abe16]. As a consequence, we obtain a proof that there exist patterns $P$ in solid tori such that $P(K)$ is not always concordant to $P(U) \# K$ and yet whose action on the smooth concordance group is invertible.