On the total absolute curvature of an immersed sphere

A recent result of Tobias Ekholm [T. Ekholm, Regular homotopy and total curvature II: Sphere immersions into 3-space, Alg. Geom. Topol. 6 (2006) 493–513] shows that for every ϵ>0ϵ>0 it is possible to construct a sphere eversion such that the total absolute curvature of the immersed spheres are always less than 8π+ϵ8π+ϵ. It is an open question whether this is the best possible. The paper contains results relating to this conjecture. As an interesting consequence of these methods it is shown that if during an eversion the total absolute curvature does not exceed 12π then a certain topological event must take place, namely the immersion must become non-simple at some point. An immersion f in general position is simple if for any irreducible self-intersection curve of f in 3-space, its two pre-image curves in the sphere are disjoint.