Algebraic Functions and Closed Braids

LET f (z,w) ≡ f0(z)w + f1(z)w + · · · + fn(z) ∈ C[z,w]. Classically, the equation f (z,w) = 0 was said to define w as an (n-valued) algebraic function of z, provided that f0(z) was not identically 0 and that f (z,w) was squarefree and without factors of the form z− c. Then, indeed, the singular set B = {z: there are not n distinct solutions w to f (z,w) = 0} is finite; and as z varies in any simply-connected domain avoiding B, the n distinct solutions w1, . . . ,wn of f (z,w) = 0 will be analytic functions of z. Now let γ be a simple closed curve in C−B. In the open solid torus γ×C ⊂ C2, the set Kγ = Vf ∩ γ×C (where Vf = {(z,w): f (z,w) = 0}) is evidently a closed 1-manifold, as smooth as γ, on which the projection to γ is an n-sheeted (possibly disconnected) covering map. A 1-manifold in a solid torus, which projects as a covering onto the circle factor, is called a closed braid. When the torus is embedded (in the standard way) in a 3-sphere (as γ×C will be, shortly), the closed braid becomes a knot or link in that sphere; if the circle factor is oriented, there is a natural way to orient that knot or link. Which such oriented links, we may ask, arise from algebraic functions (when γ is oriented counterclockwise)? The points z0 ∈ B are of two kinds (some may be of both). If, for some w0 such that f (z0,w0) = 0, it also happens that (∂ f/∂w)(z0,w0), we call z0 a singular point of the algebraic function. (Either (z0,w0) is a singular point, in the usual sense, of the algebraic curve Vf , or it is a regular point at which the tangent line is the vertical line z = z0.) At a singular point z0, some solution w to f (z0,w) = 0 has multiplicity greater than 1. On the other hand, z0 may be a root of f0(z); then there are not n solutions, even counting multiplicities, to f (z0,w) = 0. A root of f0(z) is a pole of the algebraic function. The set Kγ, being compact, actually lies in some closed solid torus γ×Dr = {(z,w): z∈ γ, |w| ≤ r}. Let B4 be the bicylinder D ×Dr where D is the bounded region in C with ∂D = γ; then B4 is homeomorphic to a 4-ball, and its boundary 3-sphere is decomposed in the usual way into two solid tori, γ×Dr and D× ∂Dr. If no pole of f (z,w) lies in D, then Kγ is the entire intersection of Vf with ∂B; that is, Vf does not meet D×∂D. (This may be seen by an appeal to the maximum modulus principle.) Below (except in §3, Remark 2) we will assume f0(z) is a (non-zero) constant, that is, that there are no poles. This is only for convenience; everything would work as well just assuming that no poles lie in D. In §2 we recall the definition of positive closed braids, and define a strictly larger class, the quasipositive closed braids. The definition is purely braid-theoretic. Several mathematicians (including Murasugi, Stallings [9], and Birman [1]) have observed that many positive closed braids, in particular all those which are knots (rather than links), are fibred links; there are quasipositive closed braids which are knots and not fibred.