I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO (see How to detect a simple closed curve from the element in the fundamental group?).
I am interested in a more specific case :
To make things simpler, let us assume we are dealing with a genus $2$ oriented closed surface with its usual set of generator of its fundamental group $a_1$, $b_1$, $ a_2$, $b_2$. It is clear that the commutator $[a_1,b_1]$ can be represented as a simple closed curve : take the curve separating $S$ in two $1$ holed torus in the classic presentation.
My question is the following : is there any other commutator on the free subgroup generated by $a_1$ and $b_1$ that can be representated by a simple closed curve ? My attempts to find such a curve all failed, checking on simple ones like $[a_1, b_1^2]$, so I wondered whether it was possible or not.