The paper deals with the problem: "Which knots or links in 3-space bound flat (immersed) compact surfaces?". In contrast to the question of `` Which knots or links in 3-space bound flat (immersed) compact surfaces?'' raised in Herman Gluck and Liu-Hua Pan, Embedding and knotting of positive curvature surfaces in 3-space, Topology, 37(4):851-873, 1998, and e.g. treated in Mohammad Ghomi's PhD thesis Strictly Convex Submanifolds And Hypersurfaces Of Positive Curvature, The Johns Hopkins University, Baltimore, Maryland, 1998, (available at http://www.math.jhu.edu/~ghomi/) the case of flat surfaces is a priory easier to experiment with due to the fact that flat surfaces are "piecewise" ruled surfaces. However, the rulings do not have to vary differentiable on a flat surface at planar points causing difficulties treating general flat surfaces. The below described experimental discovery is thus the grounding for the reported analysis of boundaries of flat compact surfaces.
On a flat surface the tangent planes are constant along a ruling. On a compact flat surface each ruling has to end at the boundary (in both directions). As the boundary curve(s) tangents lie in the shared tangent plane along the connecting ruling, the triple scalar product of these two tangents and the position vector from one ruling endpoint to the other ruling endpoint has to vanish. When dividing this triple scalar product by the length of the ruling to the third power, the integrand from the Gauss integral formula of the (self-)linking number occurs. This integrand is identical equal to zero at the diagonal. Experimenting with flat surfaces and the boundary curve differential geometry it was realized that if one instead of the third power uses the fourth power of the length of the ruling then this new function's extension to the diagonal is zero if and only is the corresponding point on the curve is a 3-singular point. A 3-singular point is either a zero curvature point or if not the a zero torsion point and generalize thus the notion vertex of a space curve.
The entire analysis of boundaries of flat compact surfaces is hereby reduced to examination of the zeros of this new function. The paper contain several examples (flat surfaces with boundary) showing that the obtained results are optimal and the paper concludes, from an experimental point of view, with a computer based conjecture that, if true, can open for transferring the used techniques to treat the problem "Which knots or links in 3-space bound flat (immersed) compact surfaces?".
Two circular Helices and a closed curve on a flat cylinder surface. The boundary curve has continuous curvature and torsion and the torsion is zero only at the two points in which it has first order contact with the rulings of the flat surface. This is the minimal number of 3-singular points (vertices) on a closed curve bonding a flat disk. The general optimal lower bound for a link to bound a flat surface is twice the absolute value of the Euler characteristic of the surface.