Computation of Rope Length of Large Thick Knots

大粗结绳索长度的计算

• 批准号: 0310562
• 负责人: Yuanan Diao
• 金额: $ 10.07万
• 依托单位: University of North Carolina at Charlotte
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0310562&HistoricalAwards=false
• 关键词: Computation; Rope; Length; Large; Thick

DMS-0310562Yuanan Diao, Claus Ernst, Uta ZieglerThis is a CARGO incubation award made under solicitation http://www.nsf.gov/pubs/2002/nsf02155/nsf02155.htm.New applications of knot theory in biophysics and biochemistry have focused interest on knotted molecules, that is, on physical knots of volume-occupying nature and particular geometric shapes. A physical knot is modeled as a smooth closed curve with certain thickness. The thickness of a smooth knot can be thought of, intuitively, as the radius of the rope with which the knot is tied as tight as possible. In mathematical terms it is the largest embedded normal tube around a smooth knot that does not intersect itself. The ropelength of a (thick) knot is the quotient of its arc length over its thickness. The focus of this proposed work is the challenging problem of finding good estimates of the minimum ropelengths of various knots. For knots with small crossing numbers, computational methods exist which estimate their ropelengths. The techniques used to obtain these estimates cannot be extended to very large knots. Thus very few computational results are available for knots with large crossing numbers. It is known that for any nontrivial knot, its minimum ropelength is bounded above by a constant times its crossing number squared. The PIs have recently improved this bound to a constant times the crossing number (of the knot) raised to the three half power. However, the PIs suspect that for most knots, their ropelengths are bounded by their crossing numbers (times a constant) or less. So there seems to be a gap between the proven bound and the actual minimal ropelengths of various knots. In this work, the PIs will develop a computer program that is capable of computing a close estimate of the minimum ropelengths of knots with large crossing numbers. This computer program will make use of the algorithm used in establishing the upper bound of the three half power mentioned above. Some research in biophysics, chemistry and physics deals with long strings of molecules that are knotted, for example, circular DNA or polymer chains. The type of knotting often influences the properties and behavior of the molecules. To better understand the properties of such molecules, this project proposes to examine the relationship between the complexity of a knot and its physical length. This relationship can be formulated into the following question: If one is given a rope of certain radius and wants to tie a certain knot, how long does the rope have to be? For small knots, one can get a rough idea by simply tying the knot and measuring how much rope it took. However for very large knots that is not practical since it is not clear how to tie a knot with rope in an optimal way. The proposed project involves the creation of a computer program that - for many knots -checks lots of different ways how the same knot can be tied, in order to find a close estimate to the minimum length of rope. The results of the computer program will be used to hypothesize a general relationship between the complexities and the lengths of knots.

DMS-0310562Yuanan Diao，Claus Ernst，Uta Ziegler这是一个根据征集而设立的CARGO孵化奖http://www.nsf.gov/pubs/2002/nsf02155/nsf02155.htm。结理论在生物物理学和生物化学中的新应用引起了人们对打结的兴趣 分子，即占据体积的物理结和特定的几何形状。物理结被建模为具有一定厚度的平滑闭合曲线。直观地，光滑结的厚度可以被认为是尽可能紧地系结的绳子的半径。用数学术语来说，它是围绕不与自身相交的光滑结的最大嵌入法线管。 （粗）结的绳长是其弧长与其厚度的商。这项工作的重点是找到各种结的最小绳长的良好估计值这一具有挑战性的问题。对于交叉数较小的结，存在估计其绳长的计算方法。用于获得这些估计的技术不能扩展到非常大的结。因此，对于具有大交叉数的结，可用的计算结果非常少。众所周知，对于任何非平凡的结，其最小绳长都以常数乘以交叉数的平方为界。 PI 最近将这个界限改进为常数乘以（结的）交叉数，提高到 3 的半次方。然而，PI 怀疑对于大多数结来说，它们的绳索长度受其交叉数（乘以常数）或更少的限制。因此，各种结的已证明的界限和实际的最小绳长之间似乎存在差距。在这项工作中，PI 将开发一个计算机程序，能够对具有大交叉数的绳结的最小绳长进行精确估计。该计算机程序将利用用于建立上述三个半幂的上限的算法。生物物理学、化学和物理学方面的一些研究涉及打结的长串分子，例如环状 DNA 或聚合物链。打结的类型通常会影响分子的性质和行为。为了更好地了解此类分子的特性，该项目建议检查结的复杂性与其物理长度之间的关系。这种关系可以表述为以下问题：如果给一个人一根一定半径的绳子，并想打一个特定的结，那么绳子必须有多长？对于小结，只需打结并测量所用绳子的长度即可大致了解。然而，对于非常大的结来说这是不切实际的，因为尚不清楚如何以最佳方式用绳子打结。拟议的项目涉及创建一个计算机程序，对于许多结，该程序检查如何打同一结的许多不同方式，以便找到对绳子最小长度的精确估计。计算机程序的结果将用于假设复杂性和结长度之间的一般关系。