Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence

$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces on $S_g\}$

Given a collection of $n$ vectors $v_1,\dots,v_n$ in $\mathbb{R}^2$ there is a natural way to construct a polygon and this polygon will represent naturally a flat surface with trivial linear holonomy (of course if I want it to be a translation surface on $S_g$ I must check that the condition on the genus by Gauss-Bonnet theorem). Then one vector $v_j$ (which is the side of the polygon between the vertices $p_j$ and $p_{j+1}$) will be the integral $\int_{p_j}^{p_{j+1}}dz=\int_{\rho_j}\omega$ where $\rho_j\in H_1(S_g,\{p_1,\dots,p_k\},\mathbb{C})$, $\{p_1,\dots,p_k\}$ singular points of $\omega$.

My question is: how do I do the opposite construction, that is how do I obtain a polygon from a couple $(X,\omega)$?

It is my understanding that it all depends in the choice of a base $\{\rho_j\}_j$ of $H_1(S_g,\{p_1,\dots,p_k\},\mathbb{C})$, then the vectors which are the sides of the polygon will be the integrals $\int_{\rho_j}\omega$. But then my question becomes: How do I determine the base $\{\rho_j\}_j$? Is it uniquely determined?

Thank you very much

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence

$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces on $S_g\}$

Given a collection of $n$ vectors $v_1,\dots,v_n$ in $\mathbb{R}^2$ there is a natural way to construct a polygon and this polygon will represent naturally a flat surface with trivial linear holonomy (of course if I want it to be a translation surface on $S_g$ I must check that the condition on the genus by Gauss-Bonnet theorem). Then one vector $v_j$ (which is the side of the polygon between the vertices $p_j$ and $p_{j+1}$) will be the integral $\int_{p_j}^{p_{j+1}}dz=\int_{\rho_j}\omega$ where $\rho_j\in H_1(S_g,\{p_1,\dots,p_k\},\mathbb{C})$, $\{p_1,\dots,p_k\}$ singular points of $\omega$.

My question is: how do I do the opposite construction, that is how do I obtain a polygon from a couple $(X,\omega)$?

It is my understanding that it all depends in the choice of a base $\{\rho_j\}_j$ of $H_1(S_g,\{p_1,\dots,p_k\},\mathbb{C})$, then the vectors which are the sides of the polygon will be the integrals $\int_{\rho_j}\omega$. But then my question becomes: How do I determine the base $\{\rho_j\}_j$? If it is always possible to obtain it, is it uniquely determined?

Thank you very much