This work brings methods from topological data analysis to knot theory and develops new data analysis tools inspired by this application. We explore a vast collection of knot invariants and relations between then using Mapper and Ball Mapper algorithms. In particular, we develop versions of the Ball Mapper algorithm that incorporate symmetries and other relations within the data, and provide ways to compare data arising from different descriptors, such as knot invariants. Additionally, we extend the Mapper construction to the case where the range of the lens function is high dimensional rather than a 1-dimensional space, that also provides ways of visualizing functions between high-dimensional spaces. We illustrate the use of these techniques on knot theory data and draw attention to potential implications of our findings in knot theory.
翻译:这项工作将从地形数据分析到结结结理论的方法, 并开发出由此应用所启发的新的数据分析工具。 我们探索了大量的结节变异性以及使用地图仪和Ball 映像仪算法的关系。 特别是, 我们开发了Ball 映像仪算法的版本, 其中包括数据中的对称和其他关系, 并提供了比较不同描述词( 如结结节变异性)产生的数据的方法。 此外, 我们将“ 映像仪” 构造扩展至镜头功能范围高维而不是一维空间, 也提供了高维空间之间功能的可视化方式。 我们用这些技术来说明结论数据, 并提请注意我们在结论中发现的潜在影响 。