Knot Theory and Complex Curves
Knot Theory and Complex Curves
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结理论和复杂曲线
DOI：
10.1090/noti2069
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发表时间：
2020
影响因子：
--
通讯作者：
Hedden, Matthew
中科院分区：
文献类型：
--
作者：
Hedden, Matthew
拓扑学经常以意想不到的美丽方式与分析和几何交织在一起。黎曼映射定理提供了这种现象的一个熟悉的例子：如果复平面上的一个域的拓扑足够简单，那么它在解析上是简单的。特别地，如果它是单连通的，并且不是完全的，那么它就完全等价于单位盘。映射定理的一种变体可以表述为“平面上没有自交的环路限定了一个与单位圆盘全纯等价的区域。”从拓扑学的角度来看，在没有自交的情况下，在2中的环是非常简单的。事实上，任何这样的循环都可以通过家庭而变形成其他任何循环。如果一个人允许循环扩展到3，情况就会变得复杂得多。结理论研究的是在没有自交的情况下，在一个由光滑变形定义的等价的环。我们可以很方便地把环看作是由紧化产生的三维球体𝕊3中的环。这个球也可以看作是在2中的单位向量的集合。从这个角度来看，我们可以问一个黎曼映射定理的类比：单位球面上有哪些结点𝕊3∧在单位4球的内部有c2界全纯嵌盘？事实证明，答案来自一个密切相关的问题的解决方案，这个问题将构成我在AMS春季中央分部会议上的演讲。回想一下，两个变量的复多项式𝑓（，𝑤）∈[，𝑤]通过它的零轨迹指定了一个代数曲线𝑓2。我的演讲将涉及以下问题。
Topology is often intertwined with analysis and geometry in unexpected and beautiful ways. A familiar example of this phenomenon is provided by the Riemann mapping theorem: if the topology of a domain in the complex plane is simple enough, then it is analytically simple. In particular, if it is simply connected and not all of ℂ, then it is holomorphically equivalent to the unit disk. A variant of the mapping theorem can be stated as “a loop in the plane without self-intersections bounds a region that is holomorphically equivalent to the unit disk.” From a topological perspective, loops in ℂ≅ ℝ2 without self-intersections are quite simple. Indeed, any such loop can be deformed to any other through a family. If one allows loops to extend into ℝ3, the situation becomes far more complicated. Knot theory studies loops in ℝ3 without self-intersections, up to an equivalence defined by smooth deformation. It is convenient to regard loops in ℝ3 as living instead in the 3-dimensional sphere 𝕊3 that arises from compactification. This sphere can also be viewed as the set of unit vectors in ℂ2. From this perspective one can ask about an analogue of the Riemann mapping theorem: which knots in the unit sphere 𝕊3⊂ ℂ2 bound holomorphically embedded disks in the interior of the unit 4-ball?It turns out that an answer comes from a solution to a closely related question that will frame my talk at the AMS Spring Central Sectional Meeting. Recall that a complex polynomial in two variables 𝑓 (𝑧, 𝑤)∈ ℂ [𝑧, 𝑤] specifies an algebraic curve 𝑉𝑓⊂ ℂ2 through its zero locus. My talk will address the following question.