Computation of Rope Length of Large Thick Knots

DMS-0310562 Diao，Claus Ernst，Uta Ziegler这是一项Cargo孵化奖，由Solation http://www.nsf.gov/pubs/2002/nsf02155/nsf02155.htm.New将纽结理论在生物物理和生物化学中的应用集中在纽结分子上，即占用体积的性质和特殊几何形状的物理纽结。物理节点被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索的半径，用来尽可能地系紧这个结。用数学术语来说，它是围绕着一个不相交的光滑结的最大嵌入法线管。(粗)结的长度是它的弧长除以它的厚度的商。这项拟议工作的重点是寻找各种结点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点，存在估计其圆周的计算方法。用来获得这些估计值的技术不能扩展到非常大的结。因此，对于交叉数较大的节点，计算结果很少。众所周知，对于任何非平凡的纽结，它的最小结点的最小长度是它的交叉数的平方的一个常数的上界。PI最近将这个界限改进到一个恒定的倍数(结的交叉数)提高到三个半次方。然而，PI怀疑对于大多数结，它们的圆周由它们的交叉数(乘以一个常量)或更少来限定。因此，在各种结点的证明边界和实际最小旋转长度之间似乎存在差距。在这项工作中，PI将开发一个计算机程序，该程序能够计算出具有较大交叉数的结点的最小距离估计。该计算机程序将利用建立上述三个半方的上界时所用的算法。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。为了更好地理解这种分子的性质，该项目建议研究结的复杂性与其物理长度之间的关系。这个关系可以表述为以下问题：如果给一个人一根半径一定的绳子，想要打一个特定的结，绳子必须有多长？对于小结，人们可以通过简单地打结并测量它需要多少绳子来得到大致的想法。然而，对于非常大的结，这是不实际的，因为它不清楚如何用绳子以一种最佳的方式打结。拟议中的项目涉及创建一个计算机程序，对于许多结，它会检查许多不同的方式如何打同一个结，以便找到一个接近最小绳长的估计。计算机程序的结果将被用来假设复杂性和结的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.