  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 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.
