Cohomology State-sum Invariants in Dimensions 3 and 4

题目：3维和4维的上同调状态和不变量：Masahiko Saito，南佛罗里达大学摘要：由主要研究者和合作者定义了三维空间中的结和四维空间中的结曲面的新的状态和不变量。选择一个有限烛光。它的元素被分配到圆弧软结图（或结面的区域）作为颜色，其中纠缠条件在每个交叉点都成立。然后，将权值以双环环的形式分配给交叉点（或三点），权值的乘积取所有交叉点（或三点），并取所有可能的颜色的和。结果表达式是状态总和变量。状态和不变量可以检测到打结曲面的不可逆性。利用有限群的量子二重上同调理论中的颜色和权值，定义了三角化4流形的相似状态和不变量。我们的项目是计算、解释和应用这些新的不变量。期望与其他理论的关系，如Seiberg-Witten不变量，量子引力的自旋泡沫模型。更高的范畴结构也研究了相关的拓扑量子场论。结是一个位于空间中的圆。结理论研究这些结环之间的差异，并应用于DNA理论和物理学。当一个结被画在一张纸上，有自己的交叉点（称为交叉点），它被称为结图。结理论中的一种方法是给结图中的弧线分配数字（称为颜色），并施加一定的规则，在交叉处分配权重，通过对所有可能的颜色取权重的总和和乘积来计算一个称为状态和的数字。状态和的概念来自统计力学。抽象的代数系统可以代替数字作为颜色。首席研究员和合作者发现了一种新的状态和，它也可以被定义为高维结点——四维空间中的打结表面。他们还发现了一个类似的四维几何物体的状态和，这些物体被分成小的四面体。该项目是计算、解释和应用这些新的状态和。这项研究需要发展对用作颜色的代数结构的复杂理解，以及对状态和性质的几何研究。期望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.