Unknotting singular surface braids by crossing changes (Q2482296)

A singular surface link is a closed oriented immersed surface \(S \subset \mathbb R^4\) whose singularities are transverse double points. Moreover, \(S\) is called a singular surface braid if it is contained in a tubular neighborhood of the standard sphere \(S^2 \subset \mathbb R^4\), in such a way that the canonical projection \(S \to S^2\) is a simple branched covering (up to desingularization). It is known that any singular surface link is isotopic to a singular surface braid. A crossing change consists in creating/deleting two opposite double points, by a finger move along an arc between different sheets of a singular surface braid, producing in this way another singular surface braid. The main result of the paper is an unknotting theorem for singular surface braids in terms of crossing changes. The proof is based on a diagrammatic argument on the chart presentation of singular surface braids. This provides a new approach to an analogous unknotting theorem for singular surface links by \textit{C. A. Giller} [Ill. J. Math. 26, 591--631 (1982; Zbl 0476.57009)]. A second result concerns the finite type invariants of singular surface braids associated with crossing changes. It says that these invariants are completely determined by the number of sheets, the Euler characteristic and the number of (signed) double points for each component.