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Surfaces in $$R^ 4$$ of braid index three are ribbon. (English) Zbl 0763.57013
A 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.

##### MSC:
 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 57Q35 Embeddings and immersions in PL-topology
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