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

Abstract

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.