Livingston's notes on concordance mention "embeddings of $S^2$ into $\mathbb{R}^4$ which are unknotted, but have non-trivial knots as cross-sections. There are other such unknotted two spheres with cross-sections that are nontrivially linked knots. And there are also pairs of spheres embedded in $\mathbb{R}^4$ which form the unlink but which have cross-sections that are nontrivial links of two components."
Could someone refer me to examples of such examples? I'm particularly interested in the conditions, necessary or sufficient, for an (pairs of) unknotted $S^2$ embedding to have a knotted (linked) cross-section.
(Just prior to that paragraph a relevant paper by E. Artin is cited, but it's in German...)
A knot is called "smoothly doubly-slice" if it is the cross-section of a smooth unknot 2-knot in $S^4$. It is called topologically doubly-slice if this trivial 2-knot is only a topologically locally-flat embedding and not necessarily a smooth embedding.
In this paper http://arxiv.org/pdf/1401.1161.pdf, Jeff Meier gave the first construction of topologically doubly slice knots which are not smoothly doubly slice.
An easy literature search returns that this notion was introduced by Fox in this article http://homepages.math.uic.edu/~kauffman/QuickTrip.pdf .The introduction to this paper http://projecteuclid.org/euclid.mmj/1029002856 provides some other sources. Searching "doubly-slice knots" should give you access to much of the modern research on the subject.