Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids

Details Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids

1998-10-15

arxiv.org/abs/math/9810095v3

Mathematics

Geometric Topology

AMS-LaTeX file; 23 pages, 14 figures; revised May 2000; to be published in Topology and Its Applications

Scientific paper

Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-plumbed if it is a 2-disk D or if it can be constructed by plumbing a positive or negative Hopf annulus A(O,-1) or A(O,1) to a Hopf-plumbed surface along a proper arc. A Seifert surface is a basket if it is D or it can be constructed by plumbing an n-twisted unknotted annulus A(O,n) to a basket along a proper arc in D. A Seifert surface is arborescent if it is D, or it is A(O,n), or it can be constructed by plumbing A(O,n) to an arborescent Seifert surface along a transverse arc of an annulus plumband. Every arborescent Seifert surface is a basket. A tree T embedded in the complex plane C determines a set of generators for a braid group. An espalier is a tree in the closed lower halfplane with vertices on the real line R. If T is an espalier then words b in the T-generators correspond nicely to T-bandword surfaces S(b). (For example, if I is an espalier with an edge from p to p+1 for p=1,...,n-1, then the I-generators of the n-string braid group are the standard generators; when Seifert's algorithm is applied to the closed braid diagram of a word in those generators, the result is an I-bandword surface.) Theorem. For any espalier T, a T-bandword surface S(b) is a Hopf-plumbed basket iff b is homogeneous iff S(b) is a fiber surface iff S(b) is connected and incompressible. A Hopf-plumbed basket S (for instance, an arborescent fiber surface) is isotopic to a homogeneous T-bandword surface for some espalier T.

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