Thurston spine in a Teichmüller curve

To study the Thurston spine ${\mathcal{P}}_g\subseteq {\mathcal{T}}_g$, we construct a Teichmüller curve $V\subseteq {\mathcal{T}}_g$. Then we characterize $V\cap {\mathcal{P}}_g$. More specifically, we show it is a trivalent tree and is an equivariant deformation retract of $V$. Moreover, by our construction, a lot of essential loops in the Thurston spine, both reducible and pseudo-Anosov, are obtained.