Einstein metrics on exotic spheres in dimensions 7, 11, and 15

In a recent article the first three authors proved that in dimension 4m + 1 all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension 4m-1, m >= 2 admit Sasakian-Einstein metrics [Boyer et al. 041, and proved this for the simplest case, namely dimension 7. In this paper we describe computer programs that show that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a program is given that determines the partition of the 8,610 deformation classes of Sasakian-Einstein metrics into the 28 distinct oriented diffornorphism types in dimension 7.