I'm trying to identify the surface $M$ constructed by gluing edges of the hexagon as shown below.
I want to take the orientability/Euler characteristic approach, but what is the general strategy when not all edges are identified?
My thought was to "straighten out" the two unidentified edges into one new edge, say $c,$ to get a pentagon with the gluing scheme $ba^{-1}a^{-1}bc.$ Then this surface is nonorientable with $$\chi(M)=V+F-E=1+1-3=-1,$$so this is a two-times connected sum of projective planes.
Is this right? If not, I would appreciate guidance as to how to fix my answer, and a general strategy to surface identification when not all edges of a polygon are included.
Thank you.