We study locally flat disks in $(\mathbb {C}P^2)^\circ :=({\mathbb {C}} P^2)\setminus \mathring {B}^4$ with boundary a fixed knot $K$ and whose complement has fundamental group $\mathbb {Z}$. We show that, up to topological isotopy relative to the (rel.) boundary, such disks necessarily arise by performing a positive crossing change on $K$ to an Alexander polynomial one knot and capping off with a $\mathbb {Z}$-disk in $D^4.$ Such a crossing change determines a loop in $S^3 \setminus K$ and we prove that the homology class of its lift to the infinite cyclic cover leads to a complete invariant of the disk. We prove that this determines a bijection between the set of rel. boundary topological isotopy classes of $\mathbb {Z}$-disks with boundary $K$ and a quotient of the set of unitary units of the ring $\mathbb {Z}[t^{\pm 1}]/(\Delta _K)$. Number-theoretic considerations allow us to deduce that a knot $K \subset S^3$ with quadratic Alexander polynomial bounds $0,1,2,4$, or infinitely many $\mathbb {Z}$-disks in $(\mathbb {C}P^2)^\circ$. This leads to the first examples of knots bounding infinitely many topologically distinct disks whose exteriors have the same fundamental group and equivariant intersection form. Finally, we give several examples where these disks are realized smoothly.

The meridional rank conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non-trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.