Research at Undergraduate Institutions Collaborative Research, Energy Functions for Knots

9420088巴克和他的同事对纽结的有限维构象空间上的能量函数所给出的能量面进行了广泛的计算和分析研究。重点讨论了两种构象空间：多边形纽结和调和纽结，其中调和纽结是由有限傅立叶级数参数化法给出的。最近的定理将这些能量函数所给出的不变量与经典的不变量联系起来，例如交叉数。因此，这个项目应该会导致对相对较低自由度可实现的结的更好理解。纽结理论是对被称为纽结的日常事物的数学研究。纽结理论曾经被认为是理论数学家的研究范围，但最近在分子生物学和宇宙学等领域得到了广泛的应用。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，研究人员将复杂的计算技术应用于结的研究。特别是，他们继续发展了一种新的打结方法：物理打结理论。数学节点被赋予物理属性，并且通过涉及这些属性的计算来区分节点类型。例如，可以给节点构象分配能量，然后可以通过考虑最小能量构象来区分节点类型。这种方法的概念基础对生物、化学、物理和数学的研究人员来说都很熟悉。该项目既产生了理论结果，也产生了软件。

9420088 Buck 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.