Cohomology State-Sum Invariants in Dimensions 3 and 4

3 维和 4 维上同调状态和不变量

• 批准号: 9988107
• 负责人: J. Scott Carter
• 金额: $ 6.73万
• 依托单位: University of South Alabama
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988107&HistoricalAwards=false
• 关键词: Cohomology; State; Sum; Invariants; Dimensions

Proposal number: 9988107Title: Cohomology state-sum invariants in dimensions 3 and 4PI: J. Scott Carter, University of South AlabamaAbstract:New state-sum invariants for knots in 3-dimensional space andknotted surfaces in 4-dimensional space are defined by theprincipal investigator and collaborators as follows.A finite quandle is chosen. Its elements are assigned to arcsof knot diagrams (or regions of knotted surfaces) as colors,where the quandle condition holds at every crossing.Weights in the form of quandle cocycles, then, are assigned tocrossings (or triple points), the product of weights are takenover all crossings (or triple points), and the sum is taken overall possible colorings. The resulting expression is the state-suminvariant. The state-sum invariant can detect non-invertibilityof knotted surfaces. Similar state-sum invariants are defined fortriangulated 4-manifolds, using colors and weights from a cohomologytheory of quantum double of finite groups. Our project is to compute,interprete, and apply these new invariants. Relations to other theories,such as Seiberg-Witten invariants, spin-foam models of quantum gravity,are expected. Higher categorical structures are also investigated inrelation to topological quantum field theories.A knot is a circle situated in space. Knot theory studies differencesamong such knotted circles, and has applications to DNA theory andphysics. When a knot is drawn on a piece of paper with self-crossingpoints (called crossings), it is called a knot diagram.One of the methods in knot theory is to assign numbers (called colors)to arcs in a knot diagram with certain rules imposed, assign weightson crossings, and compute a number called the state-sum, by takingsum and product of weights with respect to all possible colorings.The idea of state-sums came from statistical mechanics.Instead of numbers, abstract algebraic systems can be used as colors.The principal investigator and collaborators discovered a new state-sumwhich can also be defined for higher dimensional knots --- knottedsurfaces in 4-dimensional space. They also discovered a similar state-sumfor 4-dimensional geometric objects, that are divided into small 4-dimensionaltetrahedra. The project is to compute, interprete, and apply thesenew state-sums. The investigation requires developing an intricateunderstanding of the algebraic structures that are used as colors,and the geometric study of properties of the state-sums. Relations toother physical theories are expected.

建议编号：9988107标题：3维和4维上同调状态和不变量：南阿拉巴马大学J.Scott Carter摘要：主要研究者和合作者定义了3维空间中的纽结和4维空间中的纽结曲面的新的状态和不变量如下。它的元素被分配给结点图(或结点曲面的区域)的弧线作为颜色，其中在每个交叉处都存在二进制条件。然后，以二进制余圈的形式将权重分配给交叉点(或三重点)，权重的乘积被用于所有交叉点(或三重点)，并对所有可能的颜色进行求和。得到的表达式是状态和不变量。状态和不变量可以检测结点曲面的不可逆性。利用有限群量子对的上同调理论中的颜色和权重，为三角4-流形定义了类似的状态和不变量。我们的项目是计算、解释和应用这些新的不变量。与其他理论的关系，如Seiberg-Witten不变量，量子引力的自旋泡沫模型，预计会出现。我们还研究了与拓扑量子场理论相关的高级范畴结构。节点是位于空间中的圆。纽结理论研究这些打结的圆之间的差异，并将其应用于DNA理论和物理学。当在一张纸上画出具有自交叉点(称为交叉点)的纽结时，它被称为纽结图。纽结理论中的一种方法是按照一定的规则给纽结图中的弧线分配数字(称为颜色)，分配权重交叉，并通过取关于所有可能的着色的权重的和和乘积来计算称为状态和的数字。状态和的概念来自统计力学。抽象的代数系统可以用作颜色。主要研究者和合作者发现了一种新的状态和，该状态和也可以定义为4维空间中的高维纽结曲面。他们还发现了4维几何物体的类似状态和，这些物体被分成小的4维四面体。该项目是计算、解释和应用新的状态和。这项研究需要对用作颜色的代数结构进行复杂的理解，并对状态和的性质进行几何研究。与其他物理理论的联系也在意料之中。