  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.
