# Identify a curve on a closed surface of genus 4.

The notation is the one used in the attached picture.

Take a closed, orientable surface $$\Sigma_4$$ of genus $$4$$, obtained as the identification space of a polygon with $$16$$ sides in the usual way. The surface has an involution $$\iota \colon \Sigma_4 \to \Sigma_4$$, exchanging the holes $$1, \, 2$$ with the holes $$3, \, 4$$, respectively. I also have two points $$P_1$$, $$P_2$$, switched by the symmetry. In the picture, the involution is the reflection with respect to the orange curve $$c$$.

Next, let us consider the closed curves $$\rho_{11}, \, \rho_{12}, \, \tau_{11}, \, \tau_{12}$$ in the "left half" of the surface, and their image via $$\iota$$, namely $$\rho_{24}, \, \rho_{23}, \, \tau_{24}, \, \tau_{23}$$, in the "right half"; here the subscript $$ij$$ means "starting from the point $$P_i$$ and going around the $$j$$th hole" as in the picture.

Now, I would like to consider the curve $$\rho_{13}$$, colored in cyan in the figure. This curve contains $$P_1$$, crosses the mirror $$c$$ and is disjoint from $$\rho_{23}$$.

The problem is, I am unable to draw $$\rho_{13}$$ in the polygon model.

In fact, when I try to draw in the polygon a curve that starts and ends at $$P_1$$ and crosses one on the walls $$\alpha_3$$ or $$\beta_3$$, it seems to me that it intersects both $$\rho_{23}$$ and $$\tau_{23}$$. I see no way to draw such a curve so that it is disjoint from $$\rho_{23}$$.

I probably made some very basic mistakes in the identification of the curves, but I cannot see which ones. So, let me ask the

Question. What is going on? Where am I wrong, and what is the correct way to identify $$\rho_{13}$$ in the polygon model?