covering space of $2$-genus surface

Question

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one looks like $2$-genus surface which are open in one side so it's clear how to make the projection from these $2$ copies to the $2$-genus surface. my question is how can I see this process in the polygonal representation of 3-genus surface ( $12$ edges: $a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}......a_{3}b_{3}a_{3}^{-1}b_{3}^{-1})$ . I can't visualize the cut I make in the polygon. thanks.

Answer 1

Take the dodecagon at the origin with one pair of edges intersecting the $y$-axis (call them the top and bottom faces) and one pair intersecting the $x$-axis. Cut the polygon along the $x$ axis, and un-identify the left and right faces. This gives two octagons, each with an opposing pair of unmatched edges. Identify these new edges, and you should have what you're looking for.