Since the fundamental group of the Torus is $\mathbb{Z}\times \mathbb{Z}$, any homotopy class of loop in the Torus can be represented as a $(p,q)-$curve (rational slope). Am I right?
On page 3 of STEPHEN PATRIAS, he did not prove the above theorem. I checked the given reference and saw that the proof was left as an exercise. Any idea/hint on how to prove it?
I should say that the intersection number should be defined for homology classes. Now, $H_1(X) = \pi_1(X)^{\text{ab}}$, so no problems in this case.
Now, the intersection pairing is a skew-symmetric bilinear map $H_1(X)\times H_1(X) \to \mathbb{Z}$. Now, $H_1(X)$ is a free $\mathbb{Z}$ module with generators $e_1$, $e_2$. Therefore, we have $$\langle p e_1 + q e_2, p'e_1 + q' e_2 \rangle = p q' - p' q \langle e_1, e_2\rangle = ( pq' - p' q)$$
Note that if $ pq' - p' q \ne 0$, all the intersections of the curves $p e_1 + q e_2$ and $p e_1 + q e_2$ have the same orientation. Therefore, the number of points of intersection of the curves is indeed $| pq' - p' q|$.