An algebraic halfway model for the eversion of the sphere (with an appendix by Bernard Morin) (Q1813067)

An eversion of the sphere which is symmetric with respect to time, admits a halfway step (the halfway model) satisfying extra symmetric properties. A new version of the halfway model, called the closed halfway model, whose image is the set of zeros of an explicit polynomial of degree eight, is constructed. For this purpose, a 4-parameter family of halfway models is thoroughly investigated. The closed halfway model is chosen among the immersions of this family whose multiple loci contain two circles. A similar study leads to noticing that there exist Boy surfaces depending on two parameters, each of which intersects a given sphere along four circles (one parallel and three meridians). In the Appendix, Bernard Morin gives a coding in differential topological terms, of a sphere eversion which turns out to be minimal in many respects, so that, from now on, we no longer need to refer to pictures in order to present the subject.