It is well-known that in the picture below we have $t_d=(t_at_b)^6$ as elements in the mapping class group of a two-holed torus, ($t_\gamma$ represents positive Dehn twist about the curve $\gamma$). Is it possible to get another factorization of $t_d$ in terms of the curves $a$,$b$,$c$ and $e$ below?
Let's define the mapping class group to be homeomorphisms of the surface, up to isotopy. Thus a Dehn twist about a boundary component is isotopic to the identity map. Now the twist about $d$ can be expressed in terms of those about $c$, $e$, and $a$, using the lantern relation.