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The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. So they do not answer your question. But their techniques are directly relevant.

I think that you want to think about such curves in the four-holed sphere or twice-holed torus. Then, hopefully, the extra boundary components will pair up and give the desired examples in genus two.

The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. So they do not answer your question. But their techniques are directly relevant.

You should think about your curves as lying in the four-holed sphere or twice-holed torus. Then, hopefully, the extra boundary components will pair up and give the desired examples in genus two.

The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. I believe that your additional requirement turns the property from being a generic one into a very rare one.

The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. So they do not answer your question. But their techniques are directly relevant.

I think that you want to think about such curves in the four-holed sphere or twice-holed torus. Then, hopefully, the extra boundary components will pair up and give the desired examples in genus two.

The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. I believe that your additional requirement turns the property from being a generic one into a very rare one.

The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. I believe that your additional requirement turns the property from being a generic one into a very rare one.