Energy Functions for Knots

9407132 Simon研究者和他的同事对结点有限维构象空间上的能量函数给出的能量面进行了广泛的计算和分析研究。重点是两个构象空间：多边形结和调和结，其中调和结是由有限傅立叶级数参数化给出的。最近的定理将这些能量函数给出的不变量与经典不变量（如交叉数）联系起来。因此，这个项目应该导致一个更好的理解结实现相对较低的自由度。结理论是对日常生活中被称为结的物体的数学研究。曾经被认为是理论数学家的范围，结理论最近在分子生物学和宇宙学等不同领域找到了广泛的应用。例如，DNA分子通常是打结的，在复制过程中必须以某种方式解开。在这个项目中，研究人员应用复杂的计算技术来研究结。特别是，他们继续发展一种新的结的方法：物理结理论。数学结被赋予物理性质，结的类型通过涉及这些性质的计算来区分。例如，可以为结构象分配能量，然后通过考虑最小能量构象来区分结的类型。这种方法的概念基础是生物学、化学、物理学和数学研究者所熟悉的。该项目产生了理论结果和软件。

9407132 Simon The investigator and his colleague undertake an extensive computational and analytical study of the energy surfaces given by the energy functions on finite-dimensional conformation spaces of knots. The focus is on two conformation spaces: polygonal knots, and harmonic knots, where harmonic knots are those given by finite Fourier series parametrizations. Recent theorems relate the invariants given by these energy functions to classical invariants such as crossing number. So this project should lead to a better understanding of knots realizable with relatively low degrees of freedom. Knot theory is the mathematical study of the everyday objects called knots. Once thought the purview of theoretical mathematicians, knot theory has recently found a wide range of applications, in fields as diverse as molecular biology and cosmology. For example, DNA molecules are usually knotted, and in the replication process must somehow become untangled. In this project the investigators apply sophisticated computing techniques to the study of knots. In particular, they continue the development of a new approach to knots: physical knot theory. The mathematical knot is given physical properties, and knot types are distinguished by computations involving these properties. For example, knot conformations can be assigned an energy, and then knot types can be distinguished by considering minimum energy conformations. The conceptual basis of the approach is familiar to researchers in biology, chemistry, and physics as well as mathematics. The project produces both theoretical results and software.