Topological symmetries and intrinsic properties of graphs embedded in 3-space

嵌入 3 空间的图的拓扑对称性和内在属性

• 批准号: 0905087
• 负责人: Erica Flapan
• 金额: $ 18.22万
• 依托单位: Pomona College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-15 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905087&HistoricalAwards=false
• 关键词: Topological; symmetries; intrinsic; properties; graphs

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project has two goals, both concerning graphs embedded in 3-space. The topological symmetry group of a graph embedded in 3-space is defined as the group of automorphisms of the graph which are induced by homeomorphisms of 3-space. Our first goal is to characterize which groups can occur as the topological symmetry group of a particular embedded graph or family of graphs. The second goal of our project concerns intrinsic properties of a graph -- properties of an embedded graph which do not depend on the particular embedding. For example, a graph which has the property that every embedding of it in 3-space contains a nontrivial link is said to be intrinsically linked, while one which has the property that every embedding of it contains a nontrivial knot is said to be intrinsically knotted. We have previously shown that for any natural number n there is a graph such that every embedding of it contains a link with n components such that every pair of components has linking number at least n, and every component is a knot with minimal crossing number at least n. However, there cannot exist a graph which has the property that every embedding of it contains at least one composite knot. We would now like to study for which measures of complexity of knots and links there are graphs which are arbitrarily intrinsically complex by that measure.A molecule can be represented as a graph in 3-dimensional space. Most molecules are rigid, and the geometry of their graphs determines many of their properties. However, some molecules can rotate around particular bonds, and others are large enough to be somewhat flexible. For these non-rigid molecules, their topology is important in predicting their behavior. The study of the topology of graphs embedded in 3-dimensional space is used in determining the symmetries of non-rigid molecules. In particular, while the group of rigid symmetries of a molecule (known as the point group) is useful for analyzing the symmetries of rigid molecules, a molecular structure which is not rigid may have symmetries which are not included in the point group. The topological symmetry group was created in order to classify the symmetries of non-rigid molecules. The main goal of our project is to characterize all topological symmetry groups. Understanding molecular symmetries has many important application in chemistry. Symmetry is used in interpreting results in crystallography, spectroscopy, and quantum chemistry, as well as in analyzing the electron structure of a molecule. Symmetry is also used in pharmacology and in designing new molecules and new types of reactions.

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。这个项目有两个目标，都是关于嵌入3-空间的图形。嵌入三维空间的图的拓扑对称群被定义为由三维空间的同胚导出的图的自同构群。我们的第一个目标是刻画哪些群可以作为特定嵌入图或图族的拓扑对称群出现。我们项目的第二个目标涉及图的内在属性--嵌入的图的属性不依赖于特定的嵌入。例如，一个图的每个嵌入在3-空间中都包含一个非平凡链接的性质称为内在链接，而一个图的每个嵌入包含一个非平凡纽结的性质称为内在纽结。我们已经证明了对于任意自然数n，都存在一个图，使得它的每一个嵌入都包含一个有n个分支的链环，使得每一对分支至少有n个链接数，并且每个分支都是一个至少有n个最小交叉数的纽结.然而，不可能存在一个图的性质是它的每个嵌入至少包含一个复合纽结.我们现在要研究的是，对于结和环的复杂性的哪些度量，存在通过该度量而具有任意本质复杂性的图。分子可以被表示为三维空间中的图。大多数分子都是刚性的，它们的图形的几何形状决定了它们的许多性质。然而，一些分子可以围绕特定的键旋转，而另一些分子足够大，因此具有一定的灵活性。对于这些非刚性分子，它们的拓扑结构对于预测它们的行为很重要。研究嵌入在三维空间中的图的拓扑学用于确定非刚性分子的对称性。特别是，虽然分子的刚性对称性群(称为点群)对于分析刚性分子的对称性是有用的，但非刚性的分子结构可能具有不包括在点群中的对称性。为了对非刚性分子的对称性进行分类，建立了拓扑对称群。我们项目的主要目标是刻画所有的拓扑对称群。了解分子对称性在化学中有许多重要的应用。对称性用于解释结晶学、光谱学和量子化学的结果，以及分析分子的电子结构。对称性也被用于药理学和设计新的分子和新类型的反应。