Computation of Rope Length of Large Thick Knots

diyuanan Diao, Claus Ernst， Uta ziegler这是一项CARGO孵化奖http://www.nsf.gov/pubs/2002/nsf02155/nsf02155.htm.New生物物理和生物化学中结理论的应用主要集中在结分子上，即具有体积占位性质和特定几何形状的物理结。将物理结建模为具有一定厚度的光滑闭合曲线。一个光滑的结的厚度可以直观地认为是绳子的半径，绳子可以把结绑得尽可能紧。用数学术语来说，它是围绕一个光滑的结而不与自身相交的最大的嵌入正常管。（粗）结的长度是它的弧长除以它的厚度。这项工作的重点是寻找各种绳结的最小绳长的良好估计的挑战性问题。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节。因此，对于具有大交叉数的结，可用的计算结果很少。已知对于任何非平凡结，其最小绳长以常数乘以其交叉数的平方为界。pi最近将这个界限改进为一个常数乘以（结的）交叉数，提高到3 / 2次方。然而，pi怀疑，对于大多数绳结，它们的绳长是由它们的交叉数（乘以一个常数）或更小的限制的。因此，在各种绳结的被证明的界限和实际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，该程序能够计算具有大交叉数的结的最小绳长的接近估计。这个计算机程序将利用上面提到的建立三次幂的上界的算法。生物物理学、化学和物理学的一些研究涉及长串分子的打结，例如，环状DNA或聚合物链。结的类型经常影响分子的性质和行为。为了更好地理解这些分子的性质，这个项目建议研究结的复杂性和它的物理长度之间的关系。这个关系可以表述为以下问题：如果给定一根一定半径的绳子，想打一个特定的结，绳子需要有多长？对于小的结，人们可以通过简单地打结和测量需要多少绳子来得到一个大致的概念。然而，对于非常大的结，这是不实际的，因为不清楚如何以最佳的方式用绳子打结。这个项目涉及到创建一个计算机程序，对于许多绳结，该程序可以检查同一绳结的许多不同打结方式，以便找到最接近的最小绳长估计。计算机程序的结果将用于假设结的复杂性和长度之间的一般关系。

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.