Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform page-framed surgery on $L$, the resulting 3-manifold is $S^3$ since the induced open book has disk pages. Then $L$ is a fibered link in $S^3$ and we can consider the fibration $f:S^3-L\to S^1$. The question is how does $df$ look like near each component of $L$? I think we can take coordinates $(r,\theta, \phi)$ near each component (with $\theta$ representing the meridian and $\phi$ the longitude from page framing), and then $f=\theta$ and $df=d\theta$.
I am asking this question because in lemma 2.4 of this paper of Gay and Stipsicz: http://arxiv.org/abs/0908.3774, they are claiming that $df=md\theta+ld\phi$ near each component for constants $m$ and $l$ depending on the component. I cannot figure out what specific role $S^3$ is playing in their argument.
