On 2-knots with total width eight
This paper belongs to the more recent approach of studying 2-knots which was developed by the same authors in [J. Knot Theory Ramifications 18, No.~1, 41--66 (2009; Zbl 1158.57031)], and one may consider it a continuation of the mentioned paper. A 2-knot is a 2-sphere \(S^2\) smoothly embedded in \(\mathbb R^4\). The authors study 2-knots by means of generic planar projections. These projections have fold points and cusps as their singularities. Cusps appear as discrete points while fold points appear as 1-dimensional submanifolds of \(S^2\). The set of cusps and fold points in \(S^2\) form the singular set of the generic projection. The image of the singular set is called apparent contour which divides the plane into several regions and the number of regions is always even. To each such a region is associated the number of sheets covering it. The total width of a 2-knot is the minimum of the sum of these numbers, where the minimum is taken over all generic planar projections of the given 2-knot. The total width is a positive even number, and the second author has proved previously that a 2-knot has total width \(\leq 6\) if and only if it is trivial. This paper is devoted to the next total width 8. Their result says that a 2-knot has total width 8 if and only if it is an \(n\)-twist spun 2-bridge knot for some \(n\neq \pm 1\). For the proof they use the theory of braided diagrams developed by them in the above cited paper.
scientific article; zbMATH DE number 5615671
scientific article