Simple closed curves in torus.

Question

Suppose $a$ and $b$ are two simple closed curves in a one genus orientable surface, i.e the torus $K$. Assume that $a$ and $b$ intersects transversely at only one crossing in $K$. That means $a$ is not homologous to $b$ in $K$ and each is not homologous to zero in $K$. Let $l_1$, $l_2$ and $l_3$ be three simple closed homologous non-trivial curves in $K$ (non-trivial curve means it is not homologous to zero). Since these three curves are homologous in $K$, they can be represented by parallel curves in $K$. We use the notation (p,q) to represent any simple closed curve in $K$, where $(1,0)$ is the meridian and $(0,1)$ is the longitude of $K$. Suppose $a$ meets transversely each of the three curves $l_1$, $l_2$ and $l_3$ at a single crossing point. Similarly, $b$ meets transversely each of the three curves $l_1$, $l_2$ and $l_3$ at a single crossing point. I try to get the representation $(p,q)$ of $a$ and $b$. I found one possibility for the representation which is $a=(1,0)$ and $b=(1,1)$, equivalently $a=(0,1)$ and $b=(1,1)$. I wonder if there is any other possiblility for $a$ and $b$.

Answer 1

This question isn't well defined, and I don't see a reason to have three $l$ curves. If you are looking to find triples of curves that pairwise intersect in a single point, then there are infinitely many choices even after fixing one of the three curves. For example, $(1,0)$, $(n,1)$, and $(n+1,1)$. The triples of curves with this property are exactly the triangles in the Farey diagram.