It has long been known that homeomorphic 4-manifolds may admit inequivalent smooth structures. Analogous behavior holds for embedded surfaces. Some surfaces are topologically isotopic without being smoothly isotopic. Such surfaces are said to be smoothly knotted. 

It turns out that it is easier to construct inequivalent smooth structures on larger 4-manifolds. Similarly, it is easier to construct closed, smoothly knotted surfaces in large 4-manifolds. 

In this talk, we will explain how to construct smoothly knotted surfaces in a small 4-manifold. This talk will have many pictures.

This is joint work with Konno, Mukherjee, Ruberman, and Taniguchi.