algebraic intersection number of two non-trivial loops in the torus is zero

Question

Let $A$ and $B$ be two closed curves intersect on the torus transversally at a point, the intersection index of the crossing point is defined to be positive if the tangent vectors to $A$ and $B$ form an oriented basis for the tangent plane of the torus and negative otherwise. Then the intersection number of $A$ and $B$,denoted by $I(A,B)$, is the sum of the signs over all intersection points between $A$ and $B$. Suppose both $A$ and $B$ do not bound disks in the torus. Suppose also that the intersection number $I(A,B)=0$, Then does this imply that $A$ and $B$ represent the same homology class, i.e, they are homologous.

I confused because I know that the intersection number of loops in a torus can be given by

$I\big((p,q),(p',q')\big)=pq'-p'q$

Take for example $A=(5,15)$ and $B=(6,18)$ the intersection number $I(A,B)=0$, Are they homologous? I don't think so. Any guidance is highly appreciated.

Answer 1

In your example, 5 and 15 are not relatively prime, neither are 6 and 18. As a result, I think both A and B will self intersect and both are a product of (1,3) torus knots.