  While topologists have had possession of possible counterexamples to the
smooth 4-dimensional Poincar\'{e} conjecture (SPC4) for over 30 years, until
recently no invariant has existed which could potentially distinguish these
examples from the standard 4-sphere. Rasmussen's s-invariant, a slice
obstruction within the general framework of Khovanov homology, changes this
state of affairs. We studied a class of knots K for which nonzero s(K) would
yield a counterexample to SPC4. Computations are extremely costly and we had
only completed two tests for those K, with the computations showing that s was
0, when a landmark posting of Akbulut (arXiv:0907.0136) altered the terrain.
His posting, appearing only six days after our initial posting, proved that the
family of ``Cappell--Shaneson'' homotopy spheres that we had geared up to study
were in fact all standard. The method we describe remains viable but will have
to be applied to other examples. Akbulut's work makes SPC4 seem more plausible,
and in another section of this paper we explain that SPC4 is equivalent to an
appropriate generalization of Property R (``in S^3, only an unknot can yield
S^1 x S^2 under surgery''). We hope that this observation, and the rich
relations between Property R and ideas such as taut foliations, contact
geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to
look at SPC4.
