Are there explicit examples of triangulations of exotic 4spheres?

Here is my comment expanded to answer form: The question of existence of exotic 4spheres (i.e., the smooth Poincaré conjecture) is still open, and (according to Wikipedia) the existence of exotic PL structures is equivalent to it. Therefore, the answer is that no such explicit triangulations are known. In general, explicit triangulations of higher dimensional manifolds seem to be difficult to write down. I've heard from computer algebra specialists that no one has even written an explicit triangulation of $\mathbb{CP}^3$. The chaos surrounding this earlier question might suggest that the problem is subtle. 


The current status of the smooth Poincare conjecture in dimension 4 is presented in the paper: Michael Freedman, Robert Gompf, Scott Morrison and Kevin Walker "Man and machine thinking about the smooth 4dimensional Poincare conjecture" in Quantum Topology, Volume 1, Issue 2 (2010), pp. 171–208 (arXiv) Thus the CappelShaneson approach seem to fail by Akbuluts work. Now there is only possible construction via the Gluck twist with a real 2knot (i.e. knotted 2sphere), i.e. a knot not coming from a 3dimensional (classical) knot. 

