An atlas of Legendrian knots
Wutichai Chongchitmate, Lenhard Ng

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.
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