  Let $\Sigma$ be a surface with $\chi (\Sigma) < 0$, and a representation
$\rho $ from the fundamental group $\pi_1 (\Sigma)$ into $ \rm{SL} (2 ,
\mathbb{C})$. We define the \emph{trace systole} of $\rho$, denoted
$\mathrm{tys} (\rho)$ as folows : $$\mathrm{tys} (\rho) = \inf \left\{ |
\rm{tr} (\rho (\gamma)) | \ , \ \gamma \in \pi_1 (S) \mbox{ essential simple
closed curve} \right\}$$ When $\Sigma$ is endowed with an hyperbolic structure,
the trace systole of the holonomy representation is naturally related to the
usual systolic length of the hyperbolic surface, which is one of the motivation
for this study. The function $\mathrm{tys}$ is bounded on relative character
varieties of $\Sigma$, and in this article we compute explicitly the optimal
bounds for the one-holed torus, the four-holed sphere and the non-orientable
surface of genus $3$. The proofs rely on the correspondance between
representations of these surface groups and so-called Markoff maps which were
introduced by Bowditch. From this, we infer various consequences on the optimal
systolic inequalities of certain hyperbolic manifolds and also on non-Fuchsian
representations for these surfaces.
