The concordance classification of low crossing number knots

TL;DR

This paper classifies all low crossing number knots within the classical knot concordance group, providing a comprehensive understanding of their structure and extending the classification to nine crossing knots.

Contribution

It offers the first complete classification of knots with up to eight crossings in the concordance group and discusses extensions to nine crossings.

Findings

Complete classification of knots with ≤8 crossings in the concordance group

Summarized proofs of the classification

Extensions to classify nine crossing knots

Abstract

We present the complete classification of the subgroup of the classical knot concordance group generated by knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots.