Why Do We Need Two Cuts to Lay the Double-Torus Flat?

[ehrenfest]

Homework Statement

http://mathworld.wolfram.com/images/eps-gif/UniversalCoverDoubleTorus_1000.gif

Why do you need both the orange and the purple cuts to lay the double-torus flat? Why can you not connect the orange cut going right and the purple cut going left under the torus. It seems like that would be enough to lay the double-torus flat...

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

Do people understand my question?

 

[wasia]

Homework Statement

http://mathworld.wolfram.com/images/eps-gif/UniversalCoverDoubleTorus_1000.gif

Why do you need both the orange and the purple cuts to lay the double-torus flat? Why can you not connect the orange cut going right and the purple cut going left under the torus. It seems like that would be enough to lay the double-torus flat...

You are right! It is done explicitly in Introduction to algebraic curves By Phillip A. Griffiths, page 10, Figure 1.6. The book is also available on google books.