  We show that certain smooth tori with group $\mathbb{Z}$ in $S^4$ have
exteriors with standard equivariant intersection forms, and so are
topologically unknotted. These include the turned 1-twist-spun tori in the
4-sphere constructed by Boyle, the union of the genus one Seifert surface of
Cochran and Davis that has no slice derivative with a ribbon disc, and tori
with precisely four critical points whose middle level set is a 2-component
link with vanishing Alexander polynomial. This gives evidence towards the
conjecture that all $\mathbb{Z}$-surfaces in $S^4$ are topologically unknotted,
which is open for genus one and two. It is unclear whether these tori are
smoothly unknotted, except for tori with four critical points whose middle
level set is a split link. The double cover of $S^4$ branched along any of
these surfaces is a potentially exotic copy of $S^2 \times S^2$, and, in the
case of turned twisted tori, we show they cannot be distinguished from $S^2
\times S^2$ using Seiberg-Witten invariants.
