FlatKnotInfo: the first 1.24 million flat knots

TL;DR

This paper classifies and analyzes a large dataset of flat knots up to 8 crossings using various invariants, providing a searchable database to facilitate further research and pattern discovery.

Contribution

It introduces the first extensive classification of 1.24 million flat knots up to 8 crossings and provides an interactive database for researchers.

Findings

Distinguished most flat knots up to 7 crossings using multiple invariants.

Identified examples of flat knots with interesting properties like being algebraically slice but not slice.

Created a searchable online database of flat knots and their invariants.

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.