  Given a knot in $S^3$, one can associate to it a surface diffeomorphism in
two different ways. First, an arbitrary knot in $S^{3}$ can be represented by
braids, which can be thought of as diffeomorphisms of punctured disks. Second,
if the knot is fibered -- that is, if its complement fibers over $S^1$ -- one
can consider the monodromy of the fibration. One can ask to what extent
properties of these surface diffeomorphisms dictate topological properties of
the corresponding knot. In this article we collect observations, conjectures,
and questions addressing this, from both the braid perspective and the fibered
knot perspective. We particularly focus on exploring whether properties of the
surface diffeomorphisms relate to four-dimensional topological properties of
knots such as the slice genus.
