Denote the once punctured torus by $\mathbb{T}$. The fundamental group $\pi_1(\mathbb{T})$ of the once-punctured torus is the free group with two generators.
Now consider a Fuchsian representation $\rho: \pi_1(\mathbb{T}) \to \operatorname{PSL}(2, \mathbb{R})$.
To a choice of two generators $a,b$ of $\pi_1(\mathbb{T})$, we can associate two invariant geodesics $\gamma_a, \gamma_b$ on the upper-half plane $H$ corresponding to $\rho(a), \rho(b)$.
Suppose that $a', b'$ are a different choice of generators for $\pi_1(\mathbb{T})$, how are the invariant geodesics $\gamma_{a'}, \gamma_{b'}$ related (maybe as element of the free homotopy set of $H$) to $\gamma_a, \gamma_b$?
Or, I guess, a better question would be, how are $\rho(a'), \rho(b')$ related to $\rho(a), \rho(b)$ as element of $\operatorname{PSL}(2,\mathbb{R})$?