By a surface link we mean a closed oriented locally flat surface F in 4-space R~4. It is called a closed 2-dimensional braid of degree m if it is contained in a tubu- Lar neighborhood N(S~2)≌D~2×S~2 of standard 2-sphere S~2 in R~4 such that the Restriction to F of the projection D~2×S~2 → S~2 is a degree-m simple branched Covering map from F to S~2. Viro [V; cf. Ks; CS] proved that every surface link Is ambient isotopic to a closed 2-dimensional braid of degree m for some m. The Braid index of F, denoted by Braid (F), is the minimum degree among all closed 2-dimensional braids ambient isotopic to F.
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