How to construct all fibered knots and links

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A FIBERED LINK in a 3-dimensional manifold M is a collection of disjointly imbedded circles L = LI U . . . U Lk such that M - L is the total space of a fiber bundle over S’. M - L is therefore homeomorphic to F x I with ends glued by a homeomorphism h cf F. One generally requires that the fiber F meet L nicely so that h determines the pair (M, L) and the boundary of the closure F of F is L. When M = S3 this means P is a Seifert surface for L. Alexander [l] proved that every 3-manifold contains a fibered link. In the classical case M = S3 and k = 1 it was shown by Neuwirth[lO] and Stallings[l2] that a knot L is fibered if and only if the commutator subgroup of x,(M -L) is finitely generated or, equivalently, free. In the geometric vein Murasugi[9] and Stallings[l3] have given techniques for constructing fibered knots and links: The first we will use involves “plumbing” a simple fibered link of two components to a more complicated one. A sequence of plumbings increases the genus of the fiber to any needed size. The name for this operation comes from the fact that the topological type of the new fiber is that obtained by matching the two old fibers along neighborhoods of properly imbedded arcs (Fig. 3). The second construction, called “twisting,” involves a It 1 surgery on a circle C C F which is unknotted in S3. When the Seifert linking of C is correct this surgery replaces the homeomorphism h by its composition with a Dehn twist about C. The conditions guarantee a new fibered pair in S3. The roots of this operation are in[81. This connection between Dehn twists and surgery brings in the calculus of framed links [7] and allows us to prove in § l-3 our main result: THEOREM 1. Ellery fibered link in S3 is related to the unknot by a sequence of these two operations.