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.
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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?
Best Answer
There exists an infinite family of quasipositive fibre surfaces of genus 3 with the same Alexander polynomial as the torus knot T(2,7). This answers Pierre's improved question, since quasipositive surfaces do not contain any essential unlinked annuli.
Let us first recall that the fibre surface S' of T(2,7) is obtained from the fibre surface S of the torus link T(2,6) by a single positive Hopf plumbing operation. The plumbing operation is determined by a relatively embedded arc J in S. Given that the genus of S is two, there exists an infinite family of pairwise non-isotopic embedded intervals in the relative homology class of J. Plumbing positive Hopf bands along those instead of J gives rise to an infinite family of quasipositive fibred knots with the same Seifert matrix. All of them have pseudo-Anosov monodromy, since there is only one fibred knot of genus 3 with periodic monodromy of order 14: the torus knot T(2,7).