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Braided surfaces and seifert ribbons for closed braids

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Commentarii Mathematici Helvetici

Abstract

Apositive band in the braid groupB n is a conjugate of one of the standard generators; a negative band is the inverse of a positive band. Using the geometry of the configuration space, a theory of bands andbraided surfaces is developed. Each representation of a braid as a product of bands yields a handle decomposition of aSeifert ribbon bounded by the corresponding closed braid; and up to isotopy all Seifert ribbons occur in this manner. Thus,band representations provide a convenient calculus for the study of ribbon surfaces. For instance, from a band representation, a Wirtinger presentation of the fundamental group of the complement of the associated Seifert ribbon inD 4 can be immediately read off, and we recover a result of T. Yajima (and D. Johnson) that every Wirtinger-presentable group appears as such a fundamental group. In fact, we show that every such group is the fundamental group of a Stein manifold, and so that there are finite homotopy types among the Stein manifolds which cannot (by work of Morgan) be realized as smooth affine algebraic varieties.

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Authors and Affiliations

  1. Department of Mathematics, Brandeis University, 02154, Waltham, Massachusetts

    Lee Rudolph

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  1. Lee Rudolph
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Research partially supported by NSF grant MCS 76-08230

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Rudolph, L. Braided surfaces and seifert ribbons for closed braids. Commentarii Mathematici Helvetici 58, 1–37 (1983). https://doi.org/10.1007/BF02564622

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