Hyperbolic line arrangements, baskets, arborescent Seifert surfaces, generalized positive braids, and unfoldings
by
Lee Rudolph
Clark University

Let L be an arrangement of lines L₁, ..., L_m in the (real) hyperbolic plane which is transverse at infinity (i.e., for i =/= j, L_i and L_j don't share an ideal endpoint). To every weighting w:{1, ..., m} --> Z there corresponds a basket S(L, w) in S³ (i.e., a Seifert surface formed by plumbing unknotted twisted annuli to a 2-disk); conversely, every basket is isotopic to S(L, w) for some L (which may, but need not, be taken to also be transverse in the finite plane, i.e., to have no three concurrent lines) and some w. For example, the arborescent Seifert surface corresponding, in the usual way, to an evenly weighted planar tree (T, w) is a basket isotopic to S(L, -w/2) where L is a line-tree dual to T in a manner strictly analogous to the duality between ``plumbing diagrams'' and ``resolution trees'' in the theory of graph manifolds and resolution of surface singularities.

The surface S(L, w) is a fiber surface if and only if the weighting w takes values in {1, -1}, in which case S(L, w) is a Hopf-plumbed basket. In particular, for any L, the Hopf-plumbed surface S(L, -1) corresponding to the constant weighting -1 is a quasipositive Hopf-plumbed basket with an alternative representation as the braided surface corresponding to an appropriate positive word in a set of generators \sigma_e of some braid group B_n, n < m, corresponding to the edges e of an espaliered tree with vertices {1, ..., n}. For example, for n=1 (where the espaliered tree has a single edge) and k > 0, the word \sigma₁^k corresponds to a torus knot or link O{2, k} and to a hyperbolic line arrangement of type A_k-1.

In the quasipositive fibered case, at least when L is actually a euclidian line arrangement (and presumably in general, once appropriate definitions are supplied), not only does the complex line arrangement in C² obtained by complexifying L intersect every sufficiently large S³ in a fibered link of type \partialS(L, -1), but in fact the map C² --> C obtained by treating L as a divide (in a sense slightly generalizing that of A'Campo) is an unfolding (in the sense of Neumann and Rudolph) of \partialS(L, -1). The unfolding is complete if and only if L is transverse in the finite plane.

Date received: May 23, 1999