Virtual Knot Theory

虚拟结理论

基本信息

  • 批准号:
    0245588
  • 负责人:
  • 金额:
    $ 15.92万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2003
  • 资助国家:
    美国
  • 起止时间:
    2003-06-01 至 2007-05-31
  • 项目状态:
    已结题

项目摘要

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.
DMS-0245588 Louis Kauffman该项目采用了虚拟结理论的广泛方法。纽结理论研究三维空间中曲线的嵌入问题。等价纽结理论研究加厚二维球面上曲线的嵌入问题。虚纽结理论研究曲线在任意亏格的加厚曲面上的嵌入,直到在曲面上添加和移除空句柄。从经典纽结理论的观点来看,虚纽结表现为粒子的轨迹,有时从三维空间突然消失,然后在空间的另一点重新出现。这种轨迹的一个例子是一根在三维空间中运动的弦,但有时会绕道进入更高的维度。虚拟节点有一个特殊的图解理论,使得处理它们非常类似于处理经典的节点图。通过这种方法,人们可以将经典纽结理论中的许多结构推广到虚拟域,并使用虚拟纽结来测试经典问题的极限,例如琼斯多项式是否检测纽结的问题和经典庞加莱猜想。这些猜想的反例存在于虚拟域中,而这些反例中的任何一个是否等价于经典的纽结和链环(通过空句柄的加减),这是一个悬而未决的问题。虚纽结理论是一个重要的研究领域,它不仅是为了研究虚纽结理论本身,也是为了更深入地理解经典纽结理论。虚拟辫子被主要研究者用来建立量子计算、量子纠缠和拓扑纠缠之间的关系。这是一个长期的目标,这个项目的工作与结,物理学和其他自然科学,如分子生物学。推广,如虚结理论有潜力在各种各样的应用中使用,有组合的拓扑结构,组合建模的物理。在这样的应用中,拓扑结构只是图片的一部分。一个是处理系统,可以在一个离散的方式建模,以便某些特定的变化是允许的形式的模型。拓扑学家要问的问题是:什么是不变的?这一问题具有重要的应用价值,因为它涉及到分子结构的稳定性质和物理学中的守恒量。在组合拓扑学中使用的方法可以推广到各种各样的环境中使用。一个很好的例子是在分子生物学中使用纽结理论,其中允许的变化是拓扑学家认为的连续变形与对应于酶作用和重组的不连续变化的组合。这导致了结理论和分子生物学之间的有力互动。通过对纽结理论的拓扑关系和性质的研究,在分子生物学、物理学和量子计算等领域都有了新的认识和应用。

项目成果

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Louis Kauffman其他文献

Louis Kauffman的其他文献

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{{ truncateString('Louis Kauffman', 18)}}的其他基金

ICTP Summer School and Conference Knot Theory; Spring 2009, Trieste, IL
ICTP暑期学校和会议结理论;
  • 批准号:
    0925541
  • 财政年份:
    2009
  • 资助金额:
    $ 15.92万
  • 项目类别:
    Standard Grant
Polynomial Invariants in the Theory of Knots
结理论中的多项式不变量
  • 批准号:
    9802859
  • 财政年份:
    1998
  • 资助金额:
    $ 15.92万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Polynomial Invariants in the Theory of Knots
数学科学:结理论中的多项式不变量
  • 批准号:
    9504471
  • 财政年份:
    1995
  • 资助金额:
    $ 15.92万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Polynomial Invariants in the Theory of Knots
数学科学:结理论中的多项式不变量
  • 批准号:
    9205277
  • 财政年份:
    1992
  • 资助金额:
    $ 15.92万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Polynomial Invariants in Knot Theory
数学科学:结理论中的多项式不变量
  • 批准号:
    8822602
  • 财政年份:
    1989
  • 资助金额:
    $ 15.92万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Polynomial Invariants in Knot Theory
数学科学:结理论中的多项式不变量
  • 批准号:
    8701772
  • 财政年份:
    1987
  • 资助金额:
    $ 15.92万
  • 项目类别:
    Standard Grant

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