Topology and its Applications
Knot complements in S4 are characterized as follows: A connected open set W ⊂ S4 is homeomorphic to the complement of some locally flat 2-sphere in S4 if and only if H1(W) is infinite cyclic, W has one end, and the fundamental group of that end is infinite cyclic. Applications include a characterization of weakly flat 2-spheres in S4 and a complement theorem for 2-spheres in S4.
Liem, Vo Thanh and Venema, Gerard A., "Characterization of knot complements in the 4-sphere" (1991). University Faculty Publications and Creative Works. 508.