On braid monodromies of non-simple braided surfaces

A braided surface of degree m is a compact oriented surface S embedded in a bidisk such that is a branched covering map of degree m and , where is the projection. It was defined L. Rudolph [ 14, 16 ] with some applications to knot theory, cf. [ 13, 14, 15, 16, 17, 18 ]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R 4 is a closed oriented surface F embedded in R 4 such that and pr 2 │ F : F → S 2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R 4 . It is related to 2-knot theory, cf. [ 8, 9, 10 ]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [ 5, 8, 9, 14, 16 ], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [ 1 ].