标题： FlatKnotInfo：前124万种扁平纽结 显示英文标题

标题： FlatKnotInfo: the first 1.24 million flat knots

摘要： 我们使用Lyndon词上的匹配来对最多8个交叉点的平直扭结进行分类。 使用基于矩阵、$\phi$不变量、平直箭头多项式和平直琼斯-克鲁斯卡尔多项式等平直扭结不变量，我们区分了最多7个交叉点的所有平直扭结，除了五对。 在考虑的许多平直扭结中，我们找到了以下例子：(i) 代数分片但不是分片；(ii) 接近经典（同调为零）但不是分片；(iii) 非平凡但具有平凡（原始）基于矩阵。 分类数据已整理并可在FlatKnotInfo上获取，这是一个交互式可搜索的网站，列出了最多8个交叉点的平直扭结及其不变量。 它还提供了这些扭结的代数和图示信息，并设计用于使用户能够自行发现模式并提出猜想。 显示英文摘要

摘要： We use matchings on Lyndon words to classify flat knots up to 8 crossings. Using flat knots invariants such as the based matrix, the $\phi$-invariant, the flat arrow polynomial, and the flat Jones-Krushkal polynomial, we distinguish all flat knots up to 7 crossings except for five pairs. Among the many flat knots considered, we find examples that are: (i) algebraically slice but not slice; (ii) almost classical (null-homologous) but not slice; (iii) nontrivial but with trivial (primitive) based matrix. The classification data has been curated and is available on FlatKnotInfo, which is an interactive searchable website listing flat knots up to 8 crossings and their invariants. It also provides access to algebraic and diagrammatic information on these knots and is designed to enable users to discover patterns and formulate conjectures on their own.