TITLE: Genus one fibered links
QUESTION [12 upvotes]: It is well-known that the only genus one fibered knots are the trefoil and the figure-eight. On the other hand, there exist infinitely many fibered links for any fixed higher genus.
My question is about what happens if we go to more boundary components: fixing the number of boundary components, are there infinitely many genus one fibered links? 
Another related question: the monodromy of a fibered genus one link corresponds, after collapsing the boundary, to an element of $SL_n(\mathbb Z)$. For example the trefoil yields $((0,1),(-1,1))$ and the figure-eight $((2,1),(1,1))$. 
Which conjugacy classes of $SL_n(\mathbb Z)$ can be obtained as monodromies of many-components genus one fibered links?

REPLY [2 votes]: Agol's comment works to give 3 component fibered links with genus 1 fiber, but we can dial it back a bit further.   
The connect sum of a Hopf band and its mirror is a 3-component link of unknots with genus 0 fiber.  Moreover one component is 0-framed so Dehn twists along it will give an infinite family of 3 component fibered links with genus 0 fiber.  So to get an infinite family of 2 component links with genus 1 fiber, you can just plumb on another Hopf band along an arc joining the other two boundary components.  This plumbing will be disjoint from the twisting.
I think the Alexander polynomials of these links will distinguish them, perhaps even the Conway polynomials.  
It would be interesting to see the Hopf stabilization equivalences in these families.