I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to negative stabilisation).
Moreover, a stabilisation is a "hands on" operation both at the abstract level (i.e. surface with boundary and monodromy) and at the concrete level (i.e. fibred link in a three-manifold).
Whenever we have a fibred knot $K$ (say) in $S^3$, the corresponding fibration is an open book for $S^3$, which in turn supports a contact structure $\xi$ on $S^3$, where $K$ sits as a transverse knot.
Now that we have a good theoretical framework, we'd like to put our hands on some examples, and we can start stabilising the open books coming from the Hopf bands and obtain many fibred knots/links and their monodromies, and these examples are not too hard to identify as knots/links in $S^3$.
My question is purely topological: what about the inverse approach? I want to recover the monodromy on the fibre, and I have an "algorithm" to find it, once I have compressing discs for the complement of a fibre (that is a Seifert surface of minimal genus), but to me it's rather crafty. I figure that someone must have done it at some point, so:
Is there a table associating to each fibred knot the monodromy on its fibre?
I could find only one source giving concrete examples, namely Burde and Zieschang's Knots, where monodromies for the trefoil and the figure-eight knot are computed. Also, I'm pretty sure that there's much material available for algebraic knots, and that's as far as I've got.