# Characterisation of homotopy ribbon discs

## TL;DR

This paper classifies homotopy ribbon slice discs for slice knots with specific fundamental groups of their exteriors, revealing uniqueness in the infinite cyclic case and limited classes in the Baumslag-Solitar case.

## Contribution

It provides a topological classification of homotopy ribbon slice discs for certain slice knots based on their exterior's fundamental group.

## Key findings

- Unique class of such discs for infinite cyclic group
- At most two classes for Baumslag-Solitar group
- Number of discs relates to the Blanchfield pairing's lagrangians

## Abstract

Let $\Gamma$ be either the infinite cyclic group $\mathbb{Z}$ or the Baumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ be a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has fundamental group $\Gamma$. We classify the $\Gamma$-homotopy ribbon slice discs for $K$ up to topological ambient isotopy rel. boundary. In the infinite cyclic case, there is a unique equivalence class of such slice discs. When $\Gamma$ is the Baumslag-Solitar group, there are at most two equivalence classes of $\Gamma$-homotopy ribbon discs, and at most one such slice disc for each lagrangian of the Blanchfield pairing of $K$.