Mathematical Sciences: Computations for Knotted Surfaces

4维空间中的表面表现出打结行为 类似于三维空间中的圆，并且在数学上是基本的 理解四维空间主要的观点是拓扑的。 程序的基本步骤如下。(1)代表打结 表面和这些表面的同位素作为数据集。传统 拓扑结构不容易产生公式，因此 顶点坐标的计算是一件重要的事情。 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，自然的位置， 四维空间中纽结曲面的运动 第二 方法本质上是当地的，基于研究者的方法 更高维度的结移动 第三种方法包括 交互式计算机图形学 (2)表示中的打结曲面 象征性的方式。 对于四维空间中的曲面，可以将 组合对象称为结图。有一个微积分， 这种图类似于经典纽结的图 图，目标是实现这一点。 从纽结图可以看出， 确定了纽结曲面的一些代数不变量。 一 结组的呈现就是一个这样的例子。 使用实施例 （在上面的第（1）部分中产生），更高维的图将被 生成和分析，希望找到其他组合 定义不变量。(3)开发可视化技术， 搜索和验证数学关系 例子. 仅仅“看到”这些物体是不够的，人们还想得到 数学理解。 一种新的可视化技术 使用参数化纹理为以下对象提供自然的视觉提示 更高维度的信息。另一个涉及互动 在四维空间中操纵物体。 目前，调查人员正在 结合并行计算开发CAVE的使用 作为目前最好的四维互动工具， 环境 基本问题是：什么是四维空间？ 一 正在寻找数学答案，重点是计算， 视觉化 四维（或更多）的研究根本不是 深奥 从数学上讲，维度只不过是一个数字。 任何需要有四个独立变量来描述的东西 可以被认为是四维空间的子集。 最 众所周知的例子是“事件”。 一个事件发生在某个特定的 地点和时间。 需要三个数字来定位 第四个维度是时间，第四个维度是时间。的原因 四维空间与三维空间的区别在于 难以想象。 这是本研究的一个重点。 的 我们最直观的东西是表面，所以很自然地使用 四维空间中的表面作为可视化工具。在 加法曲面为四维空间提供了强有力的数学探索。 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 “超级计算机” 为了应对可视化的复杂性， 和操纵，使用“虚拟现实”系统。 在 总而言之，这一计划的广泛意义在于， 复杂信息视觉理解的前沿 现代计算工具。

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.