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?

enter image description here



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