Thurston's metric on Teichm\"uller space of semi-translation surfaces

The present paper is composed of two parts. In the first one we define two pseudo-metrics $L_F$ and $K_F$ on the Teichmu"uller space of semi-translation surfaces ${\mathcal{T}\mathcal{Q}}_g(\underset{‾}{k},ϵ)$, which are the symmetric counterparts to the metrics defined by William Thurston on ${\mathcal{T}}_g^n$. We prove some nice properties of $L_F$ and $K_F$, most notably that they are complete pseudo-metrics. In the second part we define their asymmetric analogues $L_F^a$ and $K_F^a$ on ${\mathcal{T}\mathcal{Q}}_g^{(1)}(k,ϵ)$ and prove that their equality depends on two statements regarding 1-Lipschitz maps between polygons. We are able to prove the first statement, but the second one remains a conjecture: nonetheless, we explain why we believe it is true.