发表机构
University of Southern California; Duke University（南加州大学; 杜克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文系统比较了同一光滑纽结类型的不同Legendrian代表元上的簇代数，证明了双扭结家族中这些簇结构的等价性，并刻画了Maslov-m精确Lagrangian填充的存在性。
AI 中文摘要
我们对同一光滑纽结类型的Legendrian纽结所关联的簇代数进行了系统比较。我们描述了一个包含所有可定向Lagrangian可填充扭结的双扭结家族中Legendrian代表元的m-分次增广簇上的簇结构。对于同一光滑纽结类型的不同Legendrian代表元，我们证明了这些簇结构是等价的。在此过程中，我们刻画了允许Maslov-m精确Lagrangian填充的代表元，为每个代表元构造了Catalan数个这样的填充，并证明了任何Maslov-0可填充的代表元都Legendrian同痕于一个正辫的(-1)-框架闭包。最后，我们给出了一个缆绳扭结的两个Legendrian代表元，它们产生了等价的无限型簇结构。
英文摘要
We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.
Comments46 pages, 29 figures; comments welcome!
