Cohomology State-sum Invariants in Dimensions 3 and 4

提案编号：9988101标题：3维和4维的上同调状态和不变量PI：Masahiko Saito，University of South Florida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.它的元素被分配给纽结图的弧（或纽结曲面的区域）作为颜色，其中quandle条件在每个交叉处成立。然后，将quandle上循环形式的权重分配给交叉（或三相点），在所有交叉（或三相点）上取权重的乘积，并在所有可能的着色中取总和。结果表达式是state-suminvariant。状态和不变量可以检测纽结曲面的不可逆性。类似的状态和不变量被定义为三角形的4-流形，使用颜色和重量从有限群的量子双余同调理论。我们的项目是计算，计算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。更高的范畴结构也被研究与拓扑量子场论的关系。一个纽结是一个位于空间中的圆。纽结理论研究的是这些打结的圆圈之间的差异，并将其应用于DNA理论和物理学。当一个结画在一张纸上，（称为交叉），它被称为纽结图。纽结理论中的方法之一是分配数字（称为颜色）在纽结图中的弧与某些规则强加，分配权重交叉，并计算一个数字称为状态和，通过对所有可能的着色进行加权求和和乘积。状态和的概念来自统计力学。而不是数字，抽象的代数系统可以被用作颜色。主要研究者和合作者发现了一种新的状态和，它也可以被定义为更高维的纽结--四维空间中的纽结曲面。他们还发现了类似的四维几何物体的状态和，它们被分成小的四维四面体。该项目是计算，验证和应用这些新的状态和。这项研究需要发展一个错综复杂的理解的代数结构，被用作颜色，和几何研究的属性的状态总和。与其他物理理论的关系。

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.