Real algebraic overtwisted contact structures on 3-spheres

TL;DR

This paper demonstrates that most overtwisted contact structures on the 3-sphere with positive d_3 invariant are realizable as real algebraic structures, with planar pages in their open book decompositions, expanding understanding of algebraic contact topology.

Contribution

It proves that nearly all overtwisted contact structures with positive d_3 on the 3-sphere are real algebraic, providing explicit constructions with planar open book pages.

Findings

Most overtwisted contact structures with positive d_3 are real algebraic.

Constructs open books with planar pages for these structures.

Identifies exceptions possibly related to specific invariants.

Abstract

A real algebraic link in the 3-sphere is defined as the zero locus in the 3-sphere of a real algebraic function from $\mathbb{R}^4$ to $\mathbb{R}^2$. A real algebraic open book decomposition on the 3-sphere is by definition the Milnor fibration of such a real algebraic function, in case it exists. We prove that every overtwisted contact structure on the 3-sphere with positive three dimensonal invariant $d_3$ (apart from possibly 9 exceptions) are real algebraic. Our construction shows in particular that the pages of the associated open books are planar.