用于 $s$-嵌入与 $α$-实现的立方体移动
Cube moves for $s$-embeddings and $α$-realizations
AI总结：
本文证明 $s$-嵌入中立方体移动的存在唯一性，给出共轭 Ising 星-三角变换的新耦合常数公式，并推广至 $α$-嵌入以统一调和嵌入与 $s$-嵌入。
摘要：
Chelkak 引入了 $s$-嵌入，将其作为由切圆四边形构成的铺砌，为研究任意图上具有任意耦合常数的 Ising 模型提供了合适的框架。我们证明了 $s$-嵌入的一种局部变换（称为立方体移动）的存在性与唯一性；该变换通过翻转三个四边形，使所得铺砌仍属于 $s$-嵌入类别。在此过程中，我们为 Ising 星-三角变换的耦合常数变化给出了一个新的、更简洁的公式，该变换与 $s$-嵌入的立方体移动共轭。我们更一般地引入 $α$-嵌入类别，即将平面的一部分用四边形铺砌，使每个四边形 $ABCD$ 的边长满足关系 $AB^α+CD^α=AD^α+BC^α$，从而为适用于电阻网络研究的调和嵌入（$α=2$）以及 $s$-嵌入（$α=1$）提供共同推广。我们研究了这些 $α$-嵌入的立方体移动的存在性与唯一性性质。
英文摘要：
Chelkak introduced $s$-embeddings as tilings by tangential quads which provide the right setting to study the Ising model with arbitrary coupling constants on arbitrary planar graphs. We prove the existence and uniqueness of a local transformation for $s$-embeddings called the cube move, which consists in flipping three quadrilaterals in such a way that the resulting tiling is also in the class of $s$-embeddings. In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation which is conjugated to the cube move for $s$-embeddings. We introduce more generally the class of $α$-embeddings as tilings of a portion of the plane by quadrilaterals such that the side lengths of each quadrilateral $ABCD$ satisfy the relation $AB^α+CD^α=AD^α+BC^α$, providing a common generalization for harmonic embeddings adapted to the study of resistor networks ($α=2$) and for $s$-embeddings ($α=1$). We investigate existence and uniqueness properties of the cube move for these $α$-embeddings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
