We know that cut/glue are operations in fundamental polygons in order to obtain surfaces.
Now, I wonder if you cut partially the fundamental polygon of a torus, with label word $aba^{-1}b^{-1}$, square since one of his vertex until (aproximately) its center.
So you get the middle of a diagonal. This new edge with label $c$, so the new label word is $aba^{-1}cc^{-1}b^{-1}$.
Now, can you separate this new edge to "open" the square until get a hexagon?
Thanks