Surfaces in \(R^ 4\) of braid index three are ribbon.

*(English)*Zbl 0763.57013A closed oriented surface embedded in \(\mathbb{R}^ 4\) can be described as a closed 2-dimensional braid, and its braid index defined (this concept is ascribed to O. Viro). This paper gives a method of describing a 2- dimensional braid in terms of a graph on the surface, and uses this method to show that every closed orientable surface in \(\mathbb{R}^ 4\) of braid index 3 is a ribbon surface. It is also shown that a surface has braid index 2 if and only if it is unknotted in \(\mathbb{R}^ 4\) and is homeomorphic either to a connected orientable surface of positive genus or to a pair of 2-spheres.

Reviewer: Ch.Kearton (Durham)

##### MSC:

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

57Q35 | Embeddings and immersions in PL-topology |