How to understand the simple closed curves in torus?

Question

Let $\alpha$ and $\beta$ be two simple closed curves in $T^2$ that intersect each other in one point. We identify $\alpha$ with $(1,0)\in \mathbb{Z}^2$ and $\beta$ with $(0,1)\in\mathbb{Z}^2$. Let $(p,q)$ be a primitive element of $\mathbb{Z}^2$. A simple closed curve $\gamma$ in $T^2$ is a $(p,q)$-curve if (up to sign), we have $(\hat{i}(\gamma,\beta),\hat{i}(\gamma,\alpha))=\pm(p,q).$ To construct the $(p,q)$-curve , we start by taking parallel copies of $\alpha$ and we modify this collection by a $2\pi/q$ twist along $\beta$. How to understand the last line above?

Answer 1

Each copy $\alpha_i$ of $\alpha$ intersects $\beta$ at a point $x_i.$ Cut $\alpha_i$ at $x_i,$ so you have the top end $t_i$ and the bottom end $b_i$ and connect $b_i$ to $t_{i+1}$ (where $i+1$ is taken modulo $q.$)