I have been trying to construct a 3-fold cover of the punctured torus. Specifically, I am trying to construct a once-punctured genus 2 surface. Here is an attempt. I know it's incorrect because I have a 2-fold cover that is a genus 2 surface with one boundary component, but by Euler characteristic calculations, the cover should be 3-sheeted. What is wrong with this construction, and are there any ideas for how to produce the correct cover?
