TITLE: Can we determine which monodromy of surface gives a fibered knot?
QUESTION [9 upvotes]: A fibered knot is a knot with a homeomorhism on compact surface with one boundary component. On the contrary, for a given homeomorohism on a compact surface with one boundary component, is there any way to determine whether it is a monodromy of a fibered knot?

REPLY [4 votes]: This answer gets around some of the pesky issues in Lee Mosher's answer, but it is "morally" the same his initial approach. However, this argument doesn't intrinsically use the fiberedness of the manifold, incorporating this condition might be a way to improve it. 
First, build a triangulation with one torus boundary component from the monodromy information.
Jaco and Sedgwick exhibit the $S^3-K$ decision problem (Algorithm (fancy) S in the reference below) for any cusped 3-manifold with real torus boundary (for ideal triangulations one can use Jaco-Rubinstein's inflation methods to reduce to this case). First they check that this manifold is irreducible and not a solid torus, then Jaco and Sedgwick enumerate the finitely many possible $S^3$-fillings and call upon $S^3$ recognition.  
William Jaco and Eric Sedgwick, MR 1958532 Decision problems in the space of Dehn fillings, Topology 42 (2003), no. 4, 845--906. arxiv version