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    Kamada, Seiichi and Matsumoto, Yukio 2005. Word representation of cords on a punctured plane. Topology and its Applications, Vol. 146-147, p. 21.

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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 120, Issue 2
  • August 1996, pp. 237-245

On braid monodromies of non-simple braided surfaces

  • Seiichi Kamada (a1)
  • DOI:
  • Published online: 24 October 2008

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 R4 is a closed oriented surface F embedded in R4 such that and pr2F: FS2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R4. 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].

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[1]E. Artin . Theory of braids. Ann. of Math. 48 (1947), 101126.

[2]I. Berstein and A. L. Edmonds . On the construction of branched coverings of low-dimensional manifolds. Trans. Amer. Math. Soc. 247 (1979), 87124.

[6]A. L. Edmonds , R. S. Kulkarni and R. E. Stong . Realizability of branched coverings of surfaces. Trans. Amer. Math. Soc. 282 (1984), 773790.

[7]T. Fiedler . A small state sum for knots. Topology 32 (1993), 281294.

[8]S. Kamada . Surfaces in R4 of braid index three are ribbon. J. Knot Theory Ramifications 1 (1992), 137160.

[9]S. Kamada . A characterization of groups of closed orientable surfaces in 4-space. Topology 33 (1994), 113122.

[10]S. Kamada . Alexander's and Markov's theorems in dimension four. Bull. Amer. Math. Soc. 31 (1994), 6467.

[13]L. Rudolph . Algebraic functions and closed braids. Topology 22 (1983), 191202.

[14]L. Rudolph . Braided surfaces and Seifert ribbons for closed braids. Comment. Math. Helv. 58 (1983), 137.

[15]L. Rudolph . Some topologically locally-flat surfaces in the complex projectives plane. Comment. Math. Helv. 59 (1984), 592599.

[17]L. Rudolph . A characterization of quasipositive Seifert surfaces (Constructions of quasipositive knots and links, III). Topology 31 (1992), 231237.

[18]L. Rudolph . Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. 29 (1993), 5159.

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