  Latin tableaux are a generalization of Latin squares, which first appeared in
the early 2000's in a paper of Chow, Fan, Goemans, and Vondr\'{a}k. Here, we
extend the notion of isotopy, a permutation group action, from Latin squares to
Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the
structure of orbits under the isotopy action, and investigate the relationship
between the shape of a Latin tableau and the structure of its isotopy graph.
Our main result shows that for any positive integer $d$, there is a Latin
tableau whose isotopy graph is a $d$-dimensional cube. We show that most
isotopy graphs are triangle-free, and we give a characterization of all the
Latin tableaux for which the isotopy graph contains a triangle. We also give a
formula for the degree of a vertex of each component of an isotopy graph which
depends on both the shape of the Latin Tableaux and the filling.
