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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目有两个目标，都涉及嵌入在三维空间中的图形。嵌入3-空间的图的拓扑对称群定义为由3-空间的同胚诱导的图的自同构群。我们的第一个目标是描述哪些群可以作为特定嵌入图或图族的拓扑对称群出现。我们的项目的第二个目标是关于图的内在属性-嵌入图的属性不依赖于特定的嵌入。例如，一个图如果它在3-空间中的每个嵌入都包含一个非平凡的链接，则称它是内在链接的;如果它的每个嵌入都包含一个非平凡的纽结，则称它是内在纽结的。我们已经证明，对于任意自然数n，存在一个图，使得它的每个嵌入都包含一个有n个分支的链接，使得每对分支的链接数至少为n，并且每个分支都是一个最小交叉数至少为n的纽结。然而，不可能存在一个图，它的每一个嵌入包含至少一个复合结的性质。我们现在要研究的是，对于节点和链接的复杂性的哪种度量，存在任意内在复杂的图。分子可以表示为三维空间中的图。大多数分子都是刚性的，它们的几何图形决定了它们的许多性质。然而，有些分子可以围绕特定的键旋转，而另一些分子则足够大，具有一定的柔性。对于这些非刚性分子，它们的拓扑结构在预测它们的行为方面很重要。嵌入在三维空间中的图的拓扑结构的研究用于确定非刚性分子的对称性。特别地，虽然分子的刚性对称群（称为点群）可用于分析刚性分子的对称性，但非刚性的分子结构可能具有不包括在点群中的对称性。拓扑对称群是为了对非刚性分子的对称性进行分类而建立的。我们项目的主要目标是刻画所有拓扑对称群。了解分子对称性在化学中有许多重要的应用。对称性用于解释晶体学、光谱学和量子化学的结果，以及分析分子的电子结构。对称性也用于药理学和设计新分子和新类型的反应。

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.