The Knot Spectrum of Random Knot Spaces
The Knot Spectrum of Random Knot Spaces
复制标题
随机结空间的结谱
DOI：
10.1515/9783110571493-010
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
E. Rawdon
中科院分区：
文献类型：
--
作者：
Y. Diao;C. Ernst;U. Ziegler;E. Rawdon
众所周知，自然系统中存在结。例如，在（突变体）噬菌体P4的情况下，包装在噬菌体头部内的DNA分子被认为是环状的，因为DNA的两个粘性末端彼此靠近。从衣壳中提取的DNA，不分离两端，可以保持（环形）DNA的拓扑结构，因此是明确的结。此外，在这样的系统内形成的结通常是变化的，并且不同的结以不同的概率出现。这些信息在生物学中可能很重要。在数学上，我们可以把这样一个生物系统看作一个随机的节点空间，并试图通过数学分析和数值模拟来获得关于这个系统的信息。这里的问题是找到从这个空间中随机（并且均匀）选择的结是特定结类型的概率。这相当于找到这个随机结空间内所有结类型的分布（在作者早期的论文中称为结谱）。在本文中，我们研究的行为的结谱的结多达10个交叉点。使用随机多边形的各种长度在不同的限制条件下的随机结空间（模型生物系统），我们证明了相对光谱的结，当划分成组的交叉数，仍然令人惊讶的强大，因为这些结空间的变化。对于给定的纽结类型K，我们令PK（L，R）是半径为R的限制球中长度为L的等边随机多边形具有纽结类型K的概率。我们给出了一个模型的家庭的功能PK（L，R），并表明，我们的模型函数适合我们生成的随机多边形数据。对于一个固定的交叉数Cr，3 ≤ Cr ≤ 10，令SCr是由随机多边形组成的子空间，这些多边形形成交叉数为Cr的结点。我们研究所有不同的结类型的相对分布SCr和说明如何改变这种分布，如果我们保持长度L固定（或限制半径R固定），并改变限制半径R（或长度L）。我们观察到，这种分布是相当强大的，基本上保持不变的长度和限制半径的变化，特别是如果一个集中在子家族，如交替素结，非交替素结，或复合结。Diao Yuanan，北卡罗来纳州夏洛特大学数学与统计系，夏洛特，NC 28223，USA，E-mail：ydiao@uncc.edu * 通讯作者：Claus Ernst，Department of Mathematics，Western肯塔基州大学，Bowling Green，KY 42101，USA，E-mail：乌塔·齐格勒，西肯塔基州大学工程与应用科学学院，保龄球绿色，KY 42101，USA，电子邮件：Uta.齐格勒@ wku.edu Eric J. Rawdon，数学系，University of St.托马斯，圣保罗，MN 55105，USA，电子邮件：https://doi.org/10.1515/9783110571493-009开放获取。ejrawdon@stthomas.edu wku.edu© 2018刁远安等人，由德·格鲁伊特出版。本作品采用知识共享署名-非商业性使用-禁止演绎4.0许可协议进行许可。未认证|Heruntergeladen 20.03.20 12：45 UTC 206|迪奥、恩斯特、罗登、齐格勒
It is well known that knots exist in natural systems. For example, in the case of (mutant) bacteriophage P4, DNA molecules packed inside the bacteriophage head are considered to be circular since the two sticky ends of the DNA are close to each other. The DNAs extracted from the capsid, without separating the two ends, can preserve the topology of the (circular) DNAs, and hence are well-defined knots. Furthermore, knots formed within such systems are often varied and different knots occur with different probabilities. Such information can be important in biology. Mathematically, wemay view (andmodel) such a biological system as (by) a random knot space and attempt to obtain information about the system via mathematical analysis and numerical simulation. The question here is to find the probability that a randomly (and uniformly) chosen knot from this space is of a particular knot type. This is equivalent to finding the distribution of all knot typeswithin this randomknot space (called the knot spectrum in an earlier paper by the authors). In this paper, we examine the behavior of the knot spectrums for knots up to 10 crossings. Using random polygons of various lengths under different confinement conditions as the random knot spaces (model biological systems), we demonstrate that the relative spectrums of the knots, when divided into groups by their crossing numbers, remain surprisingly robust as these knot spaces vary. For a given knot type K, we let PK(L, R) be the probability that an equilateral random polygon of length L in a confinement sphere of radius R has knot typeK. We give a model for the family of functions PK(L, R) and show that our model function fits the random polygon data we generated. For a fixed crossing number Cr, 3 ≤ Cr ≤ 10, let SCr be the subspace consisting of random polygons which form knots that have crossing number Cr. We study the relative distribution of all the different knot types within SCr and illustrate how this distribution changes if we keep the length L fixed (or the confinement radius R fixed) and vary the confinement radius R (or the length L). We observe that this distribution is quite robust and remains essentially unchanged under length and confinement radius variation, especially if one concentrates on subfamilies such as alternating prime knots, non-alternating prime knots, or composite knots. Yuanan Diao, Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA, E-mail: ydiao@uncc.edu *Corresponding author: Claus Ernst, Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA, E-mail: claus.ernst@wku.edu Uta Ziegler, School of Engineering and Applied Sciences, Western Kentucky University, Bowling Green, KY 42101, USA, E-mail: uta.ziegler@wku.edu Eric J. Rawdon, Department of Mathematics, University of St. Thomas, St. Paul, MN 55105, USA, E-mail: ejrawdon@stthomas.edu https://doi.org/10.1515/9783110571493-009 Open Access. © 2018 Yuanan Diao et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. Unauthentifiziert | Heruntergeladen 20.03.20 12:45 UTC 206 | Diao, Ernst, Rawdon, Ziegler