This paper deals with the problem: ``Which knots or links in 3-space bound flat (immersed) compact surfaces?''. In a previous paper by the author it was proven that: Any simple closed space curve can be deformed until it bounds a flat orientable compact (Seifert) surface. The main results of this paper are: There exist knots that do not bound any flat compact surfaces. The lower bound of total curvature of a knot bounding an orientable non-negatively curved compact surface can, for varying knot type, be arbitrarily much greater than the infimum of curvature needed for the knot to have its knot type. The number of $3$-singular points (points of zero curvature or if not then of zero torsion) on the boundary of a flat immersed compact surface is greater than or equal to twice the absolute value of the Euler characteristic of the surface. A set of necessary and, in a weakened sense, sufficient conditions for a knot or link to be, what we call, a generic boundary of a flat immersed compact surface without planar regions is given.
|Number of pages||30|
|Publication status||Published - 1999|
|Event||International Congress of Mathematicians - Berlin, Germany|
Duration: 18 Aug 1998 → 27 Aug 1998
|Conference||International Congress of Mathematicians|
|Period||18/08/1998 → 27/08/1998|