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: 9988101
• 负责人: Masahiko Saito
• 金额: $ 6.15万
• 依托单位: University of South Florida
• 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 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988101&HistoricalAwards=false
• 关键词: Cohomology; State; sum; Invariants; Dimensions

Proposal number: 9988101Title: Cohomology state-sum invariants in dimensions 3 and 4PI: Masahiko Saito, University of South FloridaAbstract: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.

Proposal number: 9988101Title: Cohomology state-sum invariants in dimensions 3 and 4PI: Masahiko Saito, University of South FloridaAbstract:New state-sum invariants for knots in 3-dimensional space andknotted surfaces in 4-dimensional space are defined by theprincipal investigator and collaborators如下所示。选择有限的搜索。它的元素被分配给结节图（或打结表面的区域）作为颜色，在每个交叉处都可以在这里保持乱七八糟的状态。被捕捞，所有交叉点（或三重点），并将总和总体上色。由此产生的表达是状态 - 苏姆变量。状态和状态不变可以检测到打结的表面的不可逆性。类似的状态和状态和不变性的颜色和权重定义了三角形的4个manifolds。我们的项目是计算，解释和应用这些新不变的人。与其他理论的关系，例如Seiberg-intent Infortiants，量子重力的自旋泡沫模型。还研究了较高的分类结构与拓扑量子场理论的联系。A结位于空间中。结理论研究了这种打结的圈子的差异，并在DNA理论和植物学上有应用。当打结在带有自缝点的纸上（称为交叉）时，它称为结图。结理论中的方法之一是将数字（称为颜色）分配给具有某些规则的结图。施加，分配了重量森的交叉点，并计算一个称为状态和的数字，通过takingsum和权重的产品相对于所有可能的颜色。状态和态度的想法来自统计机制。统计机制，可以使用抽象的代数系统作为颜色。他们还发现了一个类似的状态 - 阳离子4维几何对象，该物体被分为小4二维四面体。该项目是要计算，解释和应用这些国家 /地区。该研究需要对用作颜色的代数结构进行复杂的理解，并对状态和状态的性质进行几何研究。期望与其他物理理论的关系。