CDS&E: RUI: Collaborative Research: Data-Driven Methods in Classical Knot Theory

驱动这个数学研究项目的哲学是从数据驱动的角度来研究结。 该项目旨在采用强大的计算技术来大规模计算不变量，使这些算法及其输出可供社区使用，并使用计算资源来经验性地达到和测试算法。研究人员计划开发新的算法，以扩展现有的结不变量数据库，并可能导致解决旧的开放问题，从而扩大从分子生物学到量子物理学的整个科学领域的应用潜力。研究纽结的数学工具通常分为两类：几何和代数。 该项目将依靠大规模计算来分析这两个观点之间的联系，利用收集到的数据来解决迄今为止仍然难以解决的问题。 此外，该项目还打算通过开发新的组合和探索技术，使学生能够了解该领域的深层问题。 本项目的研究目标包括：（1）从计算的角度来探讨双曲秩猜想，目的是验证所有列表节点的猜想，并从这些经验发现中提取理论结果;（2）从计算的角度来研究双曲秩猜想，目的是验证所有列表节点的猜想，并从这些经验发现中提取理论结果;（2）设计一种有效的算法，通过结合从纽结图中找到这些数的上下界的技术来计算大类纽结的桥数;（3）使用Kjuchukova不变量计算切片结的同伦带状障碍，并开发算法来测试切片带状猜想的潜在反例;（4）以分支集上的奇点为结点，构造球面的分支覆盖的四维流形，对这些分支覆盖进行分类，并利用三分法研究所构造的四维流形上的光滑结构。该奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The philosophy driving this mathematics research project is to approach the study of knots from a data-driven perspective. The project aims to employ powerful computational techniques to calculate invariants on a large scale, make these algorithms and their output available to the community, and use computational resources to empirically arrive at and test conjectures. The investigators plan to develop new algorithms that will expand existing databases of knot invariants and may lead to resolving old open problems, thus also broadening the potential for applications throughout the sciences, from molecular biology to quantum physics. Mathematical tools for studying knots generally fall into two categories: geometric and algebraic. The project will rely on large-scale computations to analyze connections between these two points of view, harnessing the data collected to attack questions that have so far remained intractable. In addition, the project intends to make deep questions in the field accessible to students by developing new combinatorial and exploratory techniques. The investigators will actively engage students in impactful research, directly addressing known pipeline gaps in the field of mathematics.The research goals of this project include: (1) approaching the meridional rank conjecture from a computational perspective, with the aim of verifying the conjecture for all tabulated knots and extracting theoretical results from these empirical findings; (2) designing an efficient algorithm to compute bridge numbers for large classes of knots by combining techniques for finding lower and upper bounds for these numbers from knot diagrams; (3) computing homotopy ribbon obstructions for slice knots using Kjuchukova's invariants and developing algorithms to test potential counterexamples to the slice-ribbon conjecture; (4) constructing four-manifolds as branched covers of the sphere with knots as singularities on the branching sets; classifying these branched covers and studying, with the help of trisections, the smooth structures on the four-manifolds constructed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.