AI 中文总结
本文针对1-jet空间中勒让德曲面的增广数不变量建立纽结关系，通过二重点的扰动与拉格朗日手术等方式构造关系，并将其用于证明勒让德2-编织物增广与面着色的关系。
AI 中文摘要
对于1-jet空间中的勒让德曲面，我们研究增广数不变量，该不变量整体推广了一维勒让德纽结的多项式，并为这些不变量建立了纽结关系。主要的纽结关系涉及对具有二重点的勒让德曲面的四种消解方式：以两种不同方式将二重点扰动为可缩Reeb弦，以及通过两种拉格朗日手术移除二重点。我们在实例中应用这些纽结关系，并用它们给出Casals、Murphy和Sackel所提出的勒让德2-编织物增广与面着色之间关系的另一种证明。
英文摘要
For Legendrian surfaces in 1-jet spaces, we consider augmentation number invariants that together generalize the ruling polynomial of 1-dimensional Legendrian knots and establish skein relations for these invariants. The main skein relation involves four ways of resolving a Legendrian with a double point: perturbing the double point to a contractible Reeb chord in two different ways, and removing the double point via the two Lagrangian surgeries. We apply the skein relations in examples, and use them to provide an alternate proof of the relation between augmentations of Legendrian 2-weaves and face colorings due to Casals, Murphy, and Sackel.
Comments61 pages, 15 figures
