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。该项目旨在利用强大的计算技术来大规模地计算不变量，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地得出和检验猜想。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，并可能解决旧的开放问题，从而扩大从分子生物学到量子物理学在整个科学领域的应用潜力。研究结的数学工具一般分为两类：几何结和代数结。该项目将依靠大规模计算来分析这两种观点之间的联系，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，该项目旨在通过开发新的组合和探索技术，使学生能够接触到该领域的深层次问题。研究人员将积极地让学生参与有影响力的研究，直接解决数学领域已知的管道缺口。本项目的研究目标包括：(1)从计算角度探讨子午秩猜想，目的是验证所有表结的猜想，并从这些经验发现中提取理论结果；(2)结合从结图中寻找这些数字的下界和上界的技术，设计一种有效的算法来计算大型结的桥数；(3)利用Kjuchukova不变量计算切片结的同伦带状障碍物，并开发算法来测试切片-带状猜想的潜在反例；(4)构造四流形作为球的分支覆盖，结点作为分支集上的奇异点；对这些分支盖进行分类，并借助三分线法研究在四流形上构造的光滑结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.