three closed curves on torus

Question

Let $C_1$, $C_2$ and $C_3$ be three closed curves in a torus with the following details:

• $C_1$ is a simple close curve (that is $C_1$ has no self intersection).
• $C_1$ meets $C_2$ at a single transverse crossing point. similarly, $C_1$ meets $C_3$ at a single transverse crossing point.
• $C_2$ intersects itself at three transverse crossing points. Similarly, $C_3$ intersects itself at three transverse crossing points.
• $C_2$ and $C_3$ intersect transversely at three crossing points.

Can we find such curves in a torus or it's impossible? I think it's impossible. Because each of $C_2$ and $C_3$ intersects $C_1$ at a single transverse crossing points which implies that $C_2$ is homologous to $C_3$ (I have doubt about this statement since neither $C_2$ nor $C_3$ is a simple closed curve). So can I conclude that they are homologous?). Now $C_2$ and $C_3$ meet transversely three times so they are not homologous. This is a contradiction. Any comment about the argument above is highly appreciated.

Answer 1

It is possible. See the figure below. I think you can imagine the self-intersections of $C_2$ and $C_3$. Just insert three little loops in both curves respectively.