cubic. Since no plane intersects the twisted cubic in $4$ points, no pair of chords on the twisted cubic can cross.

In this paper, we study the following question, which might be seen as a tropical combination of these two facts.

Let $X$ be a Mumford curve over a non-Archimedean field. Does there exist a map of $X$ to a three-dimensional toric variety such that the associated tropicalization is fully faithful?

We answer this question positively, with toric variety being $(\mathbf{P}^1)^3$.

Fully and totally faithful tropicalizations
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Let us explain the analogy. Let $Y$ be a toric variety and $X$ an algebraic curve. Both $X$ and $Y$ have associated Berkovich spaces ${X^{\rm an}}$ and $Y^{\operatorname{an}}$. The toric variety $Y$ has a canonical tropicalization $\operatorname{Trop}(Y)$ which is a partial compactification of ${\mathbf{R}}^{\dim Y}$ and comes with a non-constant map $\operatorname{trop}_Y \colon Y^{\operatorname{an}} \to \operatorname{Trop}(Y)$. For a map from $\varphi \colon X \to Y$ we denote by $\operatorname{Trop}_{\varphi}(X)$ the image of the composition $\operatorname{trop}_{\varphi} \coloneqq \operatorname{trop}_Y \circ \varphi^{\operatorname{an}} \colon {X^{\rm an}}\to \operatorname{Trop}(Y)$. We call the space $\operatorname{Trop}_{\varphi}({X^{\rm an}})$ an *embedded tropical curve*. It is canonically equipped with the structure of a metric graph (potentially with edges of infinite length).

Also associated with 