Harer's conjecture on Hopf plumbing characterisation of fibered links

Let $L$ be a link in $S^3$, and let its fiber surface be the page of a fibration of its complement over $S^1$. A Hopf plumbing or deplumbing is the operation of adding or removing a positive or negative Hopf band along a suitable path on the fiber surface. Harer's conjecture. The link $L$ is fibered if and only if it and its fiber surface can be obtained from the unknot and its fiber disk via a finite sequence of successive Hopf plumbings and deplumbings.

This conjecture gives a constructive characterisation of fibered links in terms of Hopf-band operations. It was conjectured by Harer in 1982 and proved by Giroux and Goodman.