According to Thurston’s construction, which can be found for instance in Farb-Margalit’s A Primer on Mapping Class Groups, theorem 14.1 (here is a link to the version I am using: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf), If there are two curves $ a_1$ and $ a_2$ on a surface $ S_{g,n}$ (a surface of genus $ g$ with $ n$ boundary components) that fill the surface, then we will be able to figure out the type of the element (periodic, reducible, Anosov) $ \phi\in MCG(S_{g,n},\partial S_{g,n})$ , $ \phi=t_{a_2}t_{a_1}$ , based on the intersection number of $ a_1$ and $ a_2$ .
I want to know if anything can be said about the type of an element in the mapping class group that consists of three Dehn twists, say $ \psi=t_{a_3} t_{a_2} t_{a_1}$ , that $ a_1$ and $ a_2$ fill the surface and $ t_{a_3}\notin <t_{a_1},t_{a_2}>$ . Is there any general way we can determine whether such an element $ \psi$ is periodic, reducible or Anosov (perhaps based on mutual intersections of these three curves)?
PS: I am actually working on a concrete example, so that $ \psi=t_bt_at_c$ , $ \psi\in MCG(D_4,\partial D_4)$ ($ D_4$ representing a $ 4$ -puncutred disc) $ a$ and $ c$ fill the surface and $ t_at_c$ is pseudo-Anosov, but $ t_b\notin <t_a,t_c>$ as in the following figure:
