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. In this paper, I construct a new version of the halfway model for the eversion of the sphere, called the closed halfway model, whose image can readily be shown to be the set of zeros of an explicit polynomial of degree eight. For this purpose , a 4-parameter family of halfway models is thoroughly investigated. This family also contains the so-called open halfway model constructed in [A2]. The closed halfway model is chosen among the immersions of this family whose multiple loci contain two circles. Applied to the results of [A1], a similar study leads to notice 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, 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.

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