torus by identifying two equivalent points (mod $\mathbb{Z^2}$) How to visualize the quotient space $\mathbb{R^2}/ \mathbb{Z^2}$ to be a torus? 
you may also refer me to some books or websites. Because I want to see how the knot torus winds in this case.
thank you! 
 A: Think of turning a square into a donut as follows.  Start with a (stretchy) square sheet of paper (the unit square $[0,1]\times [0,1]$).  Identify the top and bottom edges to form a cylinder.  Now imagine you stretch the cylinder into a donut by bringing the two circles forming its boundary together.
Here is a nice video illustrating what I've just said.
To answer the question in your comment, think of the line $y=\frac{p}{q}x$ in the plane.  In $\mathbb{R}^2/\mathbb{Z}^2$, we are identifying all lattice points with each other.  Notice that on our line, beginning at $(0,0)$, we next reach a lattice point at $(q,p)$.  This portion of the line corresponds to the closed curve on the torus, because the portion of the line from $(q,p)$ to the next lattice point $(2q,2p)$ is just retracing the same curve.  That is, any point $(x,y)$ on this second portion is identified in $\mathbb{R}^2/\mathbb{Z}^2$ with the point $(x-q,y-p)$ which lies on the first portion.
Here is a video illustrating the case when $p=3$ and $q=4$.  As the line travels around the torus, think of the corresponding picture in the plane where we begin at $(0,0)$ traveling towards the lattice point $(4,3)$.  When we reach this point in the plane is exactly when the curve closes itself in the video.
