Characterisation of homotopy ribbon discs

Abstract

Let Γ be either the infinite cyclic group Z or the Baumslag-Solitar group Z⋉Z[12]. Let K be a slice knot admitting a slice disc D in the 4-ball whose exterior has fundamental group Γ. We classify the Γ-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 Γ is the Baumslag-Solitar group, there are at most two equivalence classes of Γ-homotopy ribbon discs, and at most one such slice disc for each lagrangian of the Blanchfield pairing of K.