基于曲线的纽结Floer同调与协变不变量综述
A curve-based survey of knot Floer homology and concordance invariants
- Department of Mathematics, Indiana University（印第安纳大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文综述了纽结Floer同调的浸入曲线解释，并从中提取协变不变量，引入广义$V_s$不变量并给出Upsilon不变量的新曲线描述。
AI中文摘要：
纽结Floer同调将$S^3$中的纽结与$\mathbb{F}[W,Z]$上的一个双分次链复形相关联，从中可以提取许多经典不变量和协变不变量。近期研究表明，这一代数对象可以等价地由标记曲面中的带装饰浸入多曲线来表示。本综述借助大量实例解释了纽结Floer同调的浸入曲线解释，并展示了如何从相应的浸入曲线中提取由纽结Floer复形产生的若干不变量。与纽结相关联的带装饰多曲线具有一个特定的曲线分量$\gamma_0$和一个特定的连通分量$\Gamma_0$，两者均为该纽结的协变不变量。我们特别关注这些分量以及可从它们提取的各种数值协变不变量。我们引入了$V_s$不变量的新推广，并通过证明Upsilon不变量由广义$V_s$不变量决定，给出了Upsilon不变量的新的基于曲线的描述。
英文摘要：
Knot Floer homology associates to a knot in $S^3$ a bigraded chain complex over $\mathbb{F}[W,Z]$, from which many classical and concordance invariants can be extracted. Recent work shows that this algebraic object can equivalently be represented by a decorated immersed multicurve in a marked surface. This survey explains the immersed curve interpretation of knot Floer homology, aided by many examples, and shows how several invariants arising from the knot Floer complex can be extracted from the corresponding immersed curves. The decorated multicurve associated to a knot has a distinguished curve component $γ_0$ and a distinguished connected component $Γ_0$, both of which are concordance invariants of the knot. We pay particular attention to these components and various numerical concordance invariants that can be extracted from them. We introduce new generalizations of the $V_s$ invariants, and we give a new curve-based description of the Upsilon invariant by showing that it is determined by generalized $V_s$ invariants.
