Characterisation of homotopy ribbon discs

Abstract

Let $\hat{I}“$ be either the infinite cyclic group $\mathbb{Z}$ or the Baumslag-Solitar group $\mathbb{Z}⋉\mathbb{Z}[\frac{1}{2}]$. Let $K$ be a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has fundamental group $\hat{I}“$. We classify the $\hat{I}“$-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 $\hat{I}“$ is the Baumslag-Solitar group, there are at most two equivalence classes of $\hat{I}“$-homotopy ribbon discs, and at most one such slice disc for each lagrangian of the Blanchfield pairing of $K$.

24 pages, 9 figures; v2: a new theorem was added (Theorem 1.6); Theorem 1.7 and Proposition 1.8 fix a statement about the number of slice discs in some of our examples v3: the new theorem from v2 (which is now Theorem 1.8) has been further improved upon; title change. v4. Further minor changes following suggestions from referees