We show that every fibred link with k components can be constructed using a simple d-sheeted cover of \(S^ 3\) branched over a suitable closed braid, with \(d=k\) for \(k\geq 3\) and otherwise \(d=3\). The method used is to relate the monodromy homeomorphism of the fibre to a homeomorphism of a disc of which it is a simple d-sheeted cover. It extends the work of Hilden and Birman, who treat the case \(k=1.\)
We go on to relate the construction of plumbing a Hopf band on to the fibre, F, of a fibred link, L, (giving a fibre for a new fibred link), with the alteration by a Markov move of the closed braid used as branch set in the covering construction for L. We show that a Markov move on the branch set always corresponds to plumbing a Hopf band to the fibre F in some way. Conversely we show how any plumbing of a Hopf band on to F can be realized by first adding trivial components to the branch set which produced L, increasing the degree of the cover correspondingly, and then conjugating the resulting braid and making a Markov move.