Boundaries of flat compact surfaces in 3-space (Q5943913)

The author handles the problem what knots or links \(\gamma\) in the Euclidean 3-space bound compact and flat immersed surfaces. For example, issuing from the total curvature \(\int_\gamma k ds\) of \(\gamma\) and applying the Gauss-Bonnet theorem, it is shown that torus knots on a thin torus of revolution do not generally bound an orientable, nonnegatively curved embedded surface. One main result (a generalized vertex theorem) is that the boundary \(\gamma\) of a compact and flat immersed surface \(S\) with Euler characteristic \(\chi(S)\) and \(p\) planar regions has at least \(2(|\chi(S) |+p)\) 3-singular points where the first three derivatives of the position vector of \(\gamma\) are linearly dependent. There exist relations of this to the four vertices theorem of \textit{V. D. Sedykh} [Bull. Lond. Math. Soc. 26, 177-180 (1994; Zbl 0807.53002)] for strictly convex closed space curves. Finally the paper, richly illustrated by many visualized examples, indicates some necessary conditions for a knot to be what the author calls ``generic'' boundary of a compact and flat immersed surface. Especially an example of a closed unknotted curve is given which does not bound any compact and developable immersed surface, a fact having importance for industrial design problems.