I am reading "Lectures on Open Book Decompositions and Contact Structures" by Etnyre. In that paper Etnyre proves existence of open books on closed oriented 3-manifolds by using two lemma's of Alexander.
First lemma of Alexander states that for any closed oriented 3-manifold M, there exists a branched cover $P:M\longrightarrow S^3$ with a branched $L_M$ where $L_M$ is a link. The other one states that any link can be braided about unknot i.e. if U denotes the unknot, taking $S^3/U\cong S^1\times\mathbb{D}^2$, $L_M$ can be isotoped such that $L_M\subset S^1\times\mathbb{D}^2$ and $L_M$ is transverse to $\{p\}\times\mathbb{D}^2$ for each $p\in S^1$.
Now comes the open book I constructed. Considering the notation given above set $B=P^{-1}(U)$. Since $P$ is a covering away from $L_M$ and U is a loop in $S^3$, by homotopy lifting, $B$ is a loop in M. Therefore one can construct an open book around B as a sequence of maps:
$\alpha_1:(B\times\mathbb{D}^2)/B\cong B\times(\mathbb{D}^2/\{0\})\longrightarrow (\mathbb{D}^2/\{0\})$ which is the projection and
$\alpha_2:\mathbb{D}^2/\{0\}\longrightarrow S^1 $ with $\alpha_2(z)=\frac{z}{|z|}$.
So we get B as the binding. But I do not understand what happens around $L_M$ living in $S^3/U$. Why do the fibers around $L_M$ have as B as the binding. Thank you for any help or suggestion.