FlatKnotInfo: the first 1.24 million flat knots

Abstract

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.