How can I judge a torus and picture's surface is homeomophic or not?

I think that image's surface is like Möbius. So not homeomorphic to sphere. How can I judge and explain about that in usual topology $$\mathbb{R}^3$$?

• Looks like an orientable surface of genus $2$ to me. – Angina Seng Oct 11 '19 at 0:37
• It's important not to get confused with how a surface is embedded iin R^3 and its homeomorphism type. – Andres Mejia Oct 11 '19 at 4:09

1 Answer

If you cut along the marked green curve, reposition the surface some, and then reglue, you can see that it is exactly a genus two surface.

• nice pictures! :) – Andres Mejia Oct 11 '19 at 4:07
• Thank you for your answer and picture. :) – rgdadl Oct 11 '19 at 5:45