Embedded surfaces with infinite cyclic knot group

We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further that imply they are ambiently isotopic. Along way, we prove certain pairs of group, homeomorphic boundaries, equivalent equivariant intersection forms, homeomorphic.