It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act periodically.
On the other hand, given a fibered knot (in $S^3$) with pseudo-Anosov monodromy, the Alexander polynomial is the characteristic polynomial of the homological monodromy, and, at the same time, its degree is twice the genus of the fiber. This implies that no monodromy of a fibered knot lies in the Torelli group.
My question is: can the homological monodromy of a knot be periodic and the geometric monodromy pseudo-Anosov? My guess is no, and this is probably known, but I could not find a reference nor a proof.
Now I think that the answer is yes, see comment below!
An improved question could then be: can the homological monodromy of a knot be periodic, the geometric monodromy be pseudo-Anosov, and the fiber surface not contain an unlinked zero-framed link?