We will discuss the knot theory of proper embeddings of the plane and half-open annulus in R4, focusing on what is arguably the simplest situation that is intrinsically noncompact. While it is still unknown if a smoothly knotted embedding S2→R4 can be topologically unknotted, there are various ways of realizing the analogous phenomenon for planes and annuli. We will show how to construct and distinguish various families of examples. These can exhibit remarkably different behavior, and can sometimes be explicitly drawn with level diagrams.