Knotted surfaces in 3-space with simple fold projections, II (Q2798948)

The author continues his investigation of knotted surfaces with simple fold projections in 3-space that he started in [J. Knot Theory Ramifications 20, No. 2, 333--342 (2011; Zbl 1215.57011)], with the goal to obtain a sufficient condition that the surface is unknotted. A surface is unknotted if the closures of the complementary regions in \(S^3\) are both handlebodies. Let \(F\) be a closed, connected and orientable surface in \(\mathbb R^3\) and \(p:\mathbb R^3\to \mathbb R^2\) a projection, such that \(p|_F:F\to \mathbb R^2\) is a generic map. \(S=S(p|_F)\) denotes the singularity set of \(p|_F\) and it consists of fold and cusp singularities, and any multiple point of \(p|_S\) is a transverse double point missing the image of cusp points. The projection is simple fold if \(p(S)\) consists of fold circles. The obtained result says: Let \(F\) be a surface in \(\mathbb R^3\) with a simple fold projection. If the fold circles of \(p(S)\) which are not innermost are concentric, then \(F\) is unknotted in \(\mathbb R^3\).