Cannot picture this surface 
This is a past exam question, but I can't picture this surface $X$ in my head. Does anyone know the name of this surface, so that I can google a 3D picture or you happen to know the equation / parametrization of it so I can plot in Mathematica ?
And for part (a), is that answer just because $X$ contain 3 mobius strips (question said: the six crossings of circles correspond to twisted bands), so it is non-orientable ?
Thanks
 A: Consider a normal vector lying in any of the shaded regions and then follow a loop from that point around one of the non-shaded regions back to the same point. You will see that the normal vector is now pointing in the other direction (it flips from 'up' to 'down' whenever you pass a crossing in the knot, which corresponds to a twist in the surface).
For the Euler characteristic just cut the surface up into discs and count vertices/edges/faces. It might help to know that you can replace a 'strip' with a twisted strip without affecting the Euler characteristic. You can also replace a twisted strip with a non-twisted strip. If you do this at all 4 crossings, you'll end up with a space which has the same Euler characteristic as the original space, but can now be embedded in the plane.
edit: a good thing to search for would be the Seifert surface of a knot/link - although keep in mind that a Seifert surface is always orientable. I found this wonderful little image on Kenn Brakke's website which I think may be the surface being described (it's hard to tell without a 3D viewer).

