Mathematical Sciences: Computations for Knotted Surfaces

四维空间中的曲面表现出与三维空间中的圆类似的打结行为，并且在数学上是基本理解四维空间的核心。主要的观点是拓扑学。程序的基本步骤如下。(1)将结状表面及其同位素表示为数据集。传统的拓扑结构不容易产生公式，因此计算顶点的坐标是一件不平凡的事情。基于结的能量概念的计算产生了有趣的，有用的，在某种意义上，在四维空间中打结表面的自然位置和运动。第二种方法是局部性质的，基于研究者的高维结移动方法。第三种方法涉及交互式计算机图形学。(2)用符号的方式表示打结的表面。我们可以将一个称为结图的组合对象与4空间中的曲面相关联。对于这样的图有一个类似于经典结图的微积分，目标是实现它。从结图可以确定结曲面的一些代数不变量。结群的展示就是这样一个例子。使用示例（在上面的第(1)部分中产生），将生成和分析高维图，希望找到其他组合定义的不变量。(3)开发可视化技术，用于搜索和验证示例之间的数学关系。仅仅“看到”这些物体是不够的，人们还需要得到数学上的理解。其中一种新的可视化技术使用参数化纹理为高维信息提供自然的视觉线索。另一个涉及4空间对象的交互式操作。目前，研究人员正在开发将CAVE与并行计算结合使用的方法，作为当前最好的工具来处理这样一个四维集成的交互环境。最基本的问题是：什么是四维空间？一个是寻找数学答案，强调计算和可视化。对四个（或更多）维度的研究一点也不深奥。在数学上，维度只不过是一个数字。任何需要四个自变量来描述的东西都可以被认为是四维空间的一个子集。最著名的例子是“事件”。一个事件发生在特定的时间和地点。需要三个数字来确定事件的位置，而“第四个维度”是时间。四维空间与三维空间大不相同的原因是它难以形象化。这是本研究的一个重点。我们视觉化的最好的东西是表面，所以使用四维空间中的表面作为视觉化工具是很自然的。此外，曲面为4空间提供了强大的数学探测。这项调查在很大程度上依赖于“最先进”的设备。为了解决计算的复杂性，使用了“超级计算机”。为了解决可视化和操作的复杂性，使用了“虚拟现实”系统。总之，该计划的广泛意义是使用现代计算工具对复杂信息的视觉理解前沿的扩展。

Surfaces in 4-dimensional space exhibit knotting behavior similar to circles in 3-space and are mathematically central to basic understanding of 4-space. The primary point of view is topological. Basic steps in the program are as follows. (1) Represent knotted surfaces and isotopies of these surfaces as data sets. Traditional topological constructions do not readily give rise to formulas, thus calculations for coordinates of vertices is a non-trivial matter. Computations based on the concept of an energy of a knot give rise to interesting, useful, and in some sense, natural placements and motions of knotted surfaces in four dimensional space. A second method is local in nature and is based on the investigator's methods of higher dimensional knot moves. A third method involves interactive computer graphics. (2) Represent knotted surfaces in a symbolic way. To a surface in 4-space one can associate a combinatorial object called a knot diagram. There is a calculus for such diagrams similar to that which exists for classical knot diagrams, a goal is to implement this. From a knot diagram one can determine some algebraic invariants of a knotted surface. A presentation of the knot group is one such example. Using examples (produced in part (1), above), higher dimensional diagrams will be generated and analyzed with the hope of finding other combinatorially defined invariants. (3) Develop visualization techniques for the search and verification of mathematical relationships between examples. It is not enough to "see" these objects, one wants to get a mathematical understanding. One such new visualization technique uses a parameterized texture to provide natural visual cues for higher dimensional information. Another involves interactive manipulations objects in 4-space. Currently, the investigator is developing use of the CAVE in conjunction with parallel computation as the best current tool for such a four-dimensional inte ractive environment. The basic question is: what is four dimensional space ? One is looking for mathematical answers with emphasis on computation and visualization. The study of four (and more) dimensions is not at all esoteric. Mathematically, a dimension is nothing more than a number. Anything that needs to have four independent variables to describe it can be considered to be a subset of four-dimensional space. The most well known example is an "event". An event happens at a certain place and a certain time. Three numbers are needed to locate the position of the event and the "fourth dimension" is time. The reason that four-dimensions is very different than three is that it is difficult to visualize. This is a focus of this research. The things that we visualize best are surfaces, so it is natural to use surfaces in four-dimensional space as a visualization tool. In addition surfaces provide powerful mathematical probes into 4-space. This investigation relies heavily on "state of the art" equipment. To grapple with the complexities of the computation, use is made of "super-computers". To grapple with the complexities of visualization and manipulation, use is made of a "virtual reality" system. In summary, the broad significance of this program is the extension of frontiers of visual understanding for complex information using modern computational tools.