On addition of 1-handles with chart loops to 2-dimensional braids (Q2833314)

The circle \(S^1\) is the only closed 1-dimensional manifold and it is the origin of knot theory in \(\mathbb R^3\). Passing to dimension 2 we have closed orientable surfaces which are the origin of knot theory in \(\mathbb R^4\) that is by today quite developed. Classical braid theory motivated the introduction of surface braids which are the subject of the present paper. Special 2-dimensional braids over a connected surface standardly embedded in \(\mathbb R^4\) have been studied by \textit{S. Hirose} [Topology Appl. 133, No. 3, 199--207 (2003; Zbl 1028.57021)] whose result is improved in this paper. The author treats 2-dimensional braids over an oriented surface knot \(F\) presented by a graph called a chart on a surface diagram of \(F\) and considers braids obtained by addition of 1-handles that are equipped with chart loops. Two such braids over \(F\) are equivalent if one is transformed into the other by ambient isotopy of \(\mathbb R^4\) whose restriction to a tubular neighborhood of \(F\) is fiber-preserving. Then moves of 1-handles with chart loops are introduced with the goal to simplify a 2-dimensional braid. A series of results is proved and we omit their description due to technical complexity. It is shown that the addition of a 1-handle with chart loops is an unbraiding operation.