Collaborative Research: Exploring the Space of Large Knots and Links

PI们建议通过理论研究和计算的共生来研究非常大的结(有数千个交叉)的空间。这里提出的问题是由于纽结理论在化学、物理和生物物理学中的应用。主要目标是开发和实现新的算法，对大节点的空间进行采样，并能够在简单的立方格中紧密或半紧密地嵌入大节点。在本研究中，理论工作将加强待开发的算法，而实证结果将支持理论方法。提出的研究项目包括几个相互关联的目标：开发能够生成大型节点图的代表性样本的快速算法，开发能够在简单立方格子中紧密嵌入大型节点的更好的算法，以及开发新的理论方法来改进一般或特殊节点类(如交替节点)的长度的上下界。这些目标是困难和具有挑战性的。例如，大结的分布是未知的，确定大非交错结的交叉数是已知的NP难问题，而关于大结的理论结果一般很少。具体地说，PI期望使用并比较三种方法来采样大节点的空间：第一，应用均匀前缀向量来生成大的哈密顿素结图；第二，采用基于均匀随机多边形的方法来采样大的素结图；最后，使用图张量积来构造大的非交替节点，其交叉数可以通过计算琼斯多项式的宽度来近似(通过Tutte多项式)；对这种非交替节点的采样允许与类似大小的交替节点进行比较。此外，PI将致力于开发更有效的嵌入算法，将其中两个PI关于闭合编织的嵌入长度的构造性证明推广到一般节点。该方法还旨在将节点长度的一般上界从目前的交叉数的上界提高到1.5的幂。这项提议要研究的主要课题是大的物理结，即在现实世界中实际可能发生的大的结。这种情况的例子有长的、打结的聚合物链或环状DNA。这项提议中提出的问题是由于纽结理论在化学、物理和生物物理学中的应用。例如，在极端的凝聚条件下形成的DNA结，比如在噬菌体P4中发现的那些，可能会非常大。如此巨大且紧密堆积的圆形DNA很难进行实验分析。对这类系统的理论结果或计算模拟将会有很大帮助。然而，关于大结的理论研究却很少。拟议中的项目旨在获得更多关于大结的知识：需要多少空间才能包装某些大结？打结的效率如何才能打得更紧？打包一个复杂的结和打包一个简单的结有区别吗？结的拓扑(形状)起什么作用？PI打算根据他们过去开发的理论结果和算法，开发能够生成大结并将其紧密包装的计算机程序。然后，可以通过重复应用这些程序来收集经验数据。拟议的活动可能会对DNA研究、聚合物科学和其他科学产生重大影响。这些活动将为研究人员提供工具，计算他们在其领域中遇到的大结的各种几何和拓扑特征，从而帮助他们更好地理解出现大结分子的生物和物理系统。

The PIs propose to investigate the space of very large knots (with thousands of crossings) through a symbiosis of theoretical research and computation. The problems raised here are motivated by the applications of knot theory in chemistry, physics, and biophysics. The main goal is to develop and implement new algorithms that sample the space of large knots and that are capable of embedding large knots tightly or semi-tightly in the simple cubic lattice. In this research, theoretical work will enhance the algorithms to be developed and the empirical results will support the theoretical approaches. The proposed research project consists of several inter-related objectives: developing fast algorithms capable of generating representative samples of large knot diagrams, developing better algorithms capable of embedding large knots tightly in the simple cubic lattice, and developing new theoretical approaches to improve the upper and lower bounds of the ropelength in general or for special knot classes such as alternating knots. These objectives are difficult and challenging. For example, the distribution of large knots is unknown, determining the crossing number of large non-alternating knots is known to be NP-hard, and theoretical results concerning large knots are scarce in general. Concretely the PIs expect to sample the space of large knots using and comparing three approaches: first applying uniform prefix vectors to generate large Hamiltonian prime knot diagrams; second, adapting a method based on uniform random polygons to sample large prime knot diagrams; finally, using graph tensor products to construct large non-alternating knots whose crossing number can be approximated by computing the breadth of the Jones polynomial (via the Tutte polynomial); sampling such non-alternating knots allows a comparison with alternating knots of similar size. Furthermore, the PIs will work on developing a more efficient embedding algorithm by extending the constructive proof of two of the PIs regarding the embedding length of closed braids to general knots. This approach also aims at improving the general upper bound on the ropelength of knots from the current bound of crossing number to the power of 1.5. The main subjects to be studied in this proposal are large physical knots, i.e. large knots that can actually occur in the real world. Examples of such occurrences are long, knotted polymer chains or circular DNA. The problems raised in this proposal are motivated by the applications of knot theory in chemistry, physics and biophysics. For example, DNA knots formed under extreme conditions of condensation, such as those found in bacteriophage P4, can be quite large. Such large and tightly packed circular DNAs are difficult to analyze experimentally. Theoretical results or computational simulations on such systems would be of great help. Yet theoretical studies on large knots are scarce. The proposed project aims at gaining more knowledge about large knots: How much space is needed in order to pack certain large knots? How efficiently can knots be packed tightly? Is there a difference between packing a complicated knot in comparison to packing a simple knot? What role does the topology (shape) of the knot play? The PIs intend to develop computer programs that can generate large knots and pack them tightly, based on theoretical results and algorithms they have developed in the past. Empirical data can then be gathered through the repeated applications of these programs. The proposed activities may have significant implications in DNA research, polymer science, and other sciences. The activities will result in tools for researchers to compute various geometric and topological characteristics of the large knots they encounter in their field and thus help them to better understand biological and physical systems where large knotted molecules occur.