The closure of a surface braid represented by a 4-chart with at most one crossing is a ribbon surface
Surface braids provide a convenient way to represent surface knots in \(\mathbb{R}^4\). In fact, \textit{S. Kamada} and \textit{O. Viro} proved that any surface knot can be isotoped to a surface braid [Topology 33, No.1, 113-122 (1994; Zbl 0820.57017)]. A surface braid of index \(n\) is a surface \(F \subset \mathbb{R}^4 - \mathbb{R}^1\) such that the restriction \(F \to S^2\) of the natural projection \(\mathbb{R}^4 - \mathbb{R}^1 \cong S^2 \times \mathbb{R}^2 \to S^2\) is a simple branched covering with \(n\) sheets. The image of \(F\) under the projection \(\mathbb{R}^4 - \mathbb{R}^1 \cong S^2 \times \mathbb{R}^2 \to S^2 \times \mathbb{R}\) is a singular surface \(F' \subset \mathbb{R}^3\). For generic \(F\), the singularities of \(F'\) can be only branch points, double curves and triple points. It is known that \(F\) is a ribbon surface, meaning that it bounds an immersed handlebody in \(\mathbb{R}^4\) with only ribbon type singularities, when \(F'\) has no triple points. Up to certain isotopy moves, this always happens for surface braids of index \(3\) [\textit{S. Kamada}, J. Knot Theory Ramifications 1, No.2, 137-160 (1992; Zbl 0763.57013)]. The proof is based on a calculus on labelled planar graphs, called \(n\)-charts, which encode both \(F\) and \(F'\). Here the authors exploit a combinatorial analysis of \(4\)-charts, in order to extend this result to the case of surface braids of index \(4\) with at most one triple point.
scientific article; zbMATH DE number 5141397
scientific article