Virtual Knot Theory

这个项目采用了广泛的方法来研究虚拟结理论。结理论研究曲线在三维空间中的嵌入。同样，结理论研究的是曲线在加厚的二维球面上的嵌入。虚拟结理论研究曲线在任意属的加厚曲面上的嵌入，直至在曲面上添加和移除空柄。从经典结理论的观点来看，虚结将以粒子的轨迹出现，粒子有时会突然从三维空间中消失，并在空间的另一点上重新出现。这种轨迹的一个例子是超弦在三维空间中运动，但偶尔会绕道进入更高的维度。虚拟结具有特殊的图解理论，使得处理它们与处理经典结图非常相似。利用这种方法，人们可以将经典结理论中的许多结构推广到虚拟域，并利用虚拟结来测试经典问题的极限，如琼斯多项式是否检测到结的问题和经典庞加莱猜想。这些猜想的反例存在于虚拟领域，这些反例是否等同于经典的结和链接（通过空句柄的加法和减法）是一个开放的问题。虚结理论是一个重要的研究领域，它本身也是对经典结理论的一个更深入的理解。首席研究员希望上述与弦理论的类比关系能结出硕果。虚拟辫子被首席研究员用来建立量子计算、量子纠缠和拓扑纠缠之间的关系。这个项目的长期目标是与结、物理学和其他自然科学（如分子生物学）合作。像虚拟结理论这样的泛化理论在各种各样的应用中都有潜在的用途，这些应用结合了拓扑和组合建模的物理性。在这样的应用程序中，拓扑结构只是全局的一部分。一种是处理可以以离散方式建模的系统，以便在模型的形式中允许某些指定的更改。拓扑学家问的问题是：在变化下什么是不变的？这个问题在应用中很重要，因为它与分子结构的稳定性质和物理学中的守恒量相对应。组合拓扑中使用的方法可以推广到各种各样的环境中。这方面的一个很好的例子是在分子生物学中使用结理论，其中允许的变化是拓扑学家认为的连续变形与与酶作用和重组相对应的不连续变化的结合。这导致了结理论和分子生物学之间强有力的相互作用。通过提出这些关于拓扑关系和结理论本质的问题，在分子生物学、物理学和量子计算方面产生了新的见解和应用。

DMS-0245588 Louis KauffmanThis project takes a broad approach to virtual knot theory. Knot theorystudies the embeddings of curves in three-dimensional space. Equivalentlyknot theory studies the embeddings of curves in athickened two dimensional sphere. Virtual knot theory studies theembeddings of curves in thickened surfaces of arbitrary genus,up to the addition and removal of empty handles from the surface. From thepoint of view of classical knot theory a virtual knotwill appear as the trajectory of a particle that sometimes abruptlydisappears from three dimensional space and reappears laterat another point in space. An example of such a trajectory would be asuperstring moving in three dimensional space, butoccasionally taking a detour into higher dimensions. Virtual knots have aspecial diagrammatic theory that makes handling themvery similar to the handling of classical knot diagrams. With thisapproach, one can generalize many structures in classical knottheory to the virtual domain, and use the virtual knots to test the limitsof classical problems such as the question whetherthe Jones polynomial detects knots and the classical Poincare conjecture.Counterexamples to these conjectures exist in thevirtual domain, and it is an open problem whether any of thesecounterexamples are equivalent (by addition and subtraction ofempty handles) to classical knots and links. Virtual knot theory is animportant domain to be investigated for its own sake and for a deeperunderstanding of classical knot theory.The principal investigator hopes that the above analog relationship with string theorywill bear fruit. Virtual braids are being used by the principalinvestigator to establish relationships among quantum computing, quantumentanglement and topological entanglement. It is a long-standing goal ofthis project to work with knots, physics andother natural sciences such as molecular biology.Generalizations such as the virtual knot theory havepotential for use in a wide variety of applications where there is acombination of topology, and combinatorially modeled physicality.In such applications, the topology is only part of the picture. One isdealing with systems that can be modeled in a discrete way sothat certain specified changes are allowed in the forms of the models. Thequestion that a topologist asks is: What is invariant under the changes?This question is significant in applications because it corresponds to thestable properties of molecular structures and to conserved quantities inthe physics. The approaches used in combinatorial topology can begeneralized for use in a wide variety of contexts. A good example of thisis seen in the use of knot theory in molecular biology where the allowedchanges are a combination of what the topologist regards as continuousdeformations coupled with discontinous changes corresponding to enzymaticaction and recombination. This has led to a vigorous interplay betweenknot theory and molecular biology. By asking these questions abouttopological relationship and the nature of knot theory, new insights andapplications in molecular biology, physics and quantum computing arecoming forth.