AI 中文总结
该研究刻画了勒让德链的分次与非分次统治多项式，构造了实现对应多项式的勒让德链，其增广簇具平凡簇代数结构，还构造了满足特定条件的勒让德纽结。
AI 中文摘要
统治多项式是与勒让德（Legendrian）的增广簇密切相关的勒让德不变量。我们刻画了所有分次与非分次统治多项式，并构造了实现每种可能多项式的勒让德链；所构造勒让德的分次增广簇均具有平凡簇代数结构。最后，我们构造了容许k个精确拉格朗日填充（χ(L)=n）且当n≤1时两两光滑非同痕、k≥0的勒让德纽结。
英文摘要
The ruling polynomial is a Legendrian invariant that is closely related to the augmentation variety of the Legendrian. We characterize all graded and ungraded ruling polynomials, and construct Legendrian links realizing each possible polynomial. The graded augmentation varieties of the Legendrians we construct all have trivial cluster algebra structures. Finally, we construct Legendrian knots admitting $k$ exact Lagrangian fillings with $χ(L)=n$ that are pairwise smoothly non-isotopic for $n\leq 1$, and $k\geq 0.$
Comments32 pages, 29 figures
