Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondrák. 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 establish that each connected component of an isotopy graph is regular, and give a formula for the degree of each vertex in a connected component of an isotopy graph which depends on both the shape of the tableau and the filling.

拉丁画面是拉丁方阵的概括，最早出现在 2000 年代初 Chow、Fan、Goemans 和 Vondrák 的论文中。在这里，我们将同位素的概念（一种置换群作用）从拉丁方格扩展到拉丁语表。我们为拉丁语表定义了同位素图，它对同位素作用下的轨道结构进行了编码，并研究了拉丁语表的形状与其同位素图的结构之间的关系。我们的主要结果表明，对于任何正整数d，都有一个拉丁表，其同位素图是d维立方体。我们展示了大多数同位素图是无三角形的，并且我们给出了同位素图包含三角形的所有拉丁语表的特征。我们还确定同位素图的每个连通分量都是规则的，并给出了同位素图连通分量中每个顶点的度数的公式，该公式取决于画面的形状和填充。