加厚环面中的全增广链环
Fully Augmented Links in the Thickened Torus
AI总结：
本文推广了 $S^3$ 中全增广链环的研究，分类了加厚环面中的双曲全增广链环，证明其补空间可分解为规范边的理想直角多面体，并证明了全增广链环的体积密度猜想。
摘要：
在本文中，我们研究了加厚环面中全增广链环补空间的几何性质，并描述了其几何特征，推广了 $S^3$ 中全增广链环的研究。我们分类了加厚环面中哪些全增广链环是双曲的，证明了它们在加厚环面中的补空间能分解为理想直角多面体，且该分解的边是规范的。我们还研究了 $S^3$ 中全增广链环的体积密度，将其定义为其体积与增广数量之比。我们证明了针对全增广链环的体积密度猜想，该猜想指出，在 $S^3$ 中图示上收敛于双周期链环的全增广链环序列，其体积密度将收敛于该双周期链环的体积密度。
英文摘要：
In this paper we study the geometry of fully augmented link complements in the thickened torus and describe their geometric properties, generalizing the study of fully augmented links in $S^3$. We classify which fully augmented links in the thickened torus are hyperbolic, show that their complements in the thickened torus decompose into ideal right-angled torihedra, and that the edges of this decomposition are canonical. We also study volume density of fully augmented links in $S^3$, defined to be the ratio of its volume and the number of augmentations. We prove the Volume Density Conjecture for fully augmented links which states that the volume density of a sequence of fully augmented links in $S^3$ which diagrammatically converge to a biperiodic link, converges to the volume density of that biperiodic link.
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