In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if R is a 2-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of R can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a 2-dimensional simply connected region is connected by flips.

在本文中，我们考虑立方体区域的所有多米诺平铺的集合。我们探索的主要问题是：我们如何才能从一种平铺移动到另一种平铺？平铺空间可以看作是具有固定度数序列的固定图的子图的空间。在代数统计中已经探索了连接这些空间的动作。因此，我们从应用代数的角度来解决这个问题，在多米诺平铺、代数统计和复曲面代数之间建立新的联系。使用图的复曲面理想的结果，我们能够描述连接任何维度的给定立方体区域的平铺空间的移动。这是通过研究由同一区域的两个不同多米诺瓦片产生的二项式来完成的。此外，我们介绍了平铺理想和翻转理想并使用这些理想重述平铺空间翻转连接的含义。最后，我们证明如果R是一个二维单连通立方区域，任何由R 的两个不同平铺产生的二项式都可以用二次二项式来写。作为我们主要结果的推论，我们获得了一个替代证据，证明二维单连接区域的多米诺骨牌拼贴集是通过翻转连接的。