An atlas of Legendrian knots

TL;DR

This paper provides a comprehensive atlas of Legendrian knots in standard contact three-space, offering a conjectural classification for knots with low arc index, and introduces new phenomena and infinite families of transversely nonsimple knots.

Contribution

It presents the first extensive atlas of Legendrian knots, including a computer-assisted classification method and new examples of complex knot phenomena.

Findings

Conjectural classification for all knots with arc index ≤ 9.

Identification of new phenomena like transverse nonsimplicity.

Discovery of infinite families of transversely nonsimple knots.

Abstract

We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots through 7 crossings and nonalternating knots through 9 crossings. Our method involves a computer search of grid diagrams and applies to transverse knots as well. The atlas incorporates a number of new, small examples of phenomena such as transverse nonsimplicity and non-maximal non-destabilizable Legendrian knots, and gives rise to new infinite families of transversely nonsimple knots.