Three-dimensional braids and their descriptions (Q897435)

A 3-dimensional braid is a generalization of the notion of braid into three dimensions. Let \(D^2\) and \(B^3\) be a 2-disk and a 3-disk respectively. A 3-dimensional braid in \(D^2 \times B^3\) of degree \(d\) is a 3-manifold \(M\) embedded in \(D^2 \times B^3\) such that (i) the restriction map \(pr_2|_M: M \to B^3\) is a simple branched covering map of degree \(d\) branched along a link in \(B^3\) and (ii) \(\partial M=M \cap \partial (D^2 \times B^3)=X_d \times \partial B^3\), where \(pr_2: D^2 \times B^3 \to B^3\) is the second factor projection, and \(X_d\) is a fixed set of \(d\) interior points of \(D^2\). In this paper, the authors study 3-dimensional braids and show that 3-dimensional braids are described by braid monodromies or curtains. A curtain is a certain 2-complex in a 3-ball \(B^3\) such that each face is oriented and labeled in \(\{1,\dots, d-1\}\), where \(d\) is the degree of the 3-dimensional braid. The authors introduce the notion of a curtain, and show how to construct a curtain for any 3-dimensional braid.