Narrow Results

We define a type of Stallings twists, which represents a "complexity" of the twist, and show that the Stallings twists of a certain type on a fiber surface for a fibered link in S³ can be realized by plumbing one Hopf band and deplumbing another Hopf band. Using this technique, we construct stable Hopf plumbings which are not Hopf plumbings with an arbitrary high first Betti number.

We give a recurrence formula for calculating the Alexander polynomials of 2-bridge links by using a special type of Conway diagram. As an application we give sufficient and necessary conditions for the periodic covering link over a 2-bridge link to be a fibered link. We also give a recurrence formula to calculate the reduced Alexander polynomials of periodic covering links over 2-bridge links.

We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer’s twisting, which is presented in two different (but stably equivalent) forms. Our approach relies on 4-dimensional Lefschetz fibrations, and on 3-dimensional contact topology, via the Giroux-Goodman stable equivalence theorem for open book decompositions representing homologous contact structures.

We use the Birman–Ko–Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element $\delta$ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of $B_n$ and we prove that Rudolph’s condition is equivalent to ours. Finally, we show that the braid index is a strict upper bound for the number of crossing changes required to fiber a strongly quasipositive braid.

In this paper, we show that every closed orientable surface bundle over the circle is represented by a fibered link in the 3-sphere with framings induced by the fibration of the complement.