Virtual Knot Theory

DMS-0245588 Louis Kauffman这个项目采用了一种广泛的方法来研究虚拟结理论。纽结理论研究曲线在三维空间中的嵌入。等价纽结理论研究了曲线在固定的二维球面上的嵌入问题。虚结点理论研究了曲线在任意亏格的加厚曲面上的嵌入，直至从曲面上添加和移除空手柄。从经典纽结理论的观点来看，虚拟纽结将作为粒子的轨迹出现，有时会突然从三维空间消失，然后在空间的另一个点重新出现。这种轨迹的一个例子是一根在三维空间中运动的超弦，但它临时绕道进入了更高的维度。虚拟结有一种特殊的图解理论，使它们的处理非常类似于经典结图的处理。利用这种方法，我们可以将经典纽结理论中的许多结构推广到虚域上，并利用虚结来检验经典问题的极限，例如琼斯多项式是否检测到纽结和经典的庞加莱猜想。这些猜想的反例在虚域中存在，而这些例子是否等同于经典的纽结和链环(通过空句柄的加减)是一个悬而未决的问题。虚结理论是一个重要的研究领域，对于它本身和对经典纽结理论的深入理解，主要研究者希望上述与弦理论的类比关系能够取得成果。首席调查员正在使用虚拟辫子来建立量子计算、量子纠缠和拓扑纠缠之间的关系。这个项目的长期目标是研究纽结、物理学和其他自然科学，如分子生物学。像虚拟纽结理论这样的泛化理论有潜力用于各种应用，其中存在拓扑学和组合建模物理的组合。在这样的应用中，拓扑学只是画面的一部分。一种是处理可以以离散方式建模的系统，以便允许在模型的形式中进行某些特定的更改。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，因为它对应于分子结构的稳定性质和物理学中的守恒量。组合拓扑学中使用的方法可以推广到广泛的上下文中使用。分子生物学中纽结理论的使用就是一个很好的例子，在分子生物学中，允许的变化是拓扑学家认为的连续变形加上与酶作用和重组相对应的不连续变化的组合。这导致了纽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纽结理论本质的问题，人们在分子生物学、物理学和量子计算方面有了新的见解和应用。

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.