operator. And, as we show below, this is indeed true.

The infinitesimal monodromy and Glick’s operator {#Sec:proof}
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In this section we prove our main result, Theorem \[thm1\] (=Theorem \[thm2\]). To that end, we explicitly construct a deformation $P_z$ of a polygon $P$ as in Definition \[def:im\]. Such a deformation is not unique, but we know that the infinitesimal monodromy does not depend on the deformation. We will in fact use this ambiguity to our advantage by choosing a deformation for which the infinitesimal monodromy can be computed explicitly. We will then compute it and see that it coincides with Glick’s operator.

Consider a closed $n$-gon $P$. Lift the $n$-periodic sequence $\{v_i \in {\mathbb P}^2\}$ of its vertices to an $n$-periodic sequence of non-zero vectors $V_i \in {\mathbb R}^3$. Then, for every $i \in {\mathbb Z}$, there exist $a_i, b_i, c_i \in {\mathbb R}$ such that $$\begin{aligned}
\label{eq4}
    V_{i+3}=a_i V_{i+2} + b_i V_{i+1} + c_i V_i. \end{aligned}$$ Furthermore, for a generic polygon the numbers $a_i, b_i, c_i$ are uniquely determined because the points $v_{i}, v_{i+1}, v_{i+2}$ are not collinear so the vectors $V_i, V_{i+1}, V_{i+3}$ are linearly independent. Also, we have $c_i \neq 0$ for any $i$ because the points 