Legendrian knot and link atlases

On this page, there are links to atlases of Legendrian knots and two-component links of topological type with arc index at most 9. Various source files are also available, including Java code for anyone who might be interested in extending the atlas or otherwise playing with grid diagrams for Legendrian knots.

These files accompany the paper “An atlas of Legendrian knots” by Wutichai Chongchitmate and Lenhard Ng, arXiv:1010.3997, which is based on Wutichai Chongchitmate's undergraduate senior thesis, also available below. Special thanks also to the 2014-2015 edition of DRM@Duke, whose participants produced the latest Java programs.

Recent changes:

• May 2011: added new data about transverse nonsimplicity derived from the transverse homology invariant.
• August 2013: corrected the ruling polynomial for one of the m(10₁₆₁) knots (thanks to Aaron Lowe for catching this).
• October 2015: updated m(9₄₅) with the information from Augmentations are sheaves, section 4.4.5, which shows that the second diagram for m(9₄₅) is not Legendrian isotopic to its mirror.
• March 2024: updated m(9₄₄) to correct an error discovered by Carlo Collari and Paolo Lisca.
  

The Legendrian knot atlas
This is the main atlas, with explanation, as a standalone file. Previous version: October 2015 Legendrian knot atlas.

Atlas for Legendrian two-component links

Legend/explanation for the Legendrian two-component link atlas

Classification of Legendrian knots and links by Wutichai Chongchitmate: senior thesis, Duke University, 2010

Source files:

• XO Permutation.pdf, XO Permutation.tex: a table of the grid diagrams depicted in the knot atlas, represented by their X-O data. For knots with multiple grid diagrams, the order of the grid diagrams agrees with the order that they appear in the knot atlas.
• braid presentation.pdf, braid presentation.tex: a table of the Legendrian knots corresponding to the grid diagrams in the knot atlas, expressed as plat closures of braids. This may be useful for input into the Mathematica program “Legendrian invariants.nb”, available here, which computes various Legendrian knot invariants. Note: for starred knot types, plat presentations are not guaranteed to appear in the order that the corresponding grid diagrams appear in the atlas.

• atlas-code.tar: the Java programs used to produce the Legendrian knot atlas, written by Wutichai Chongchitmate. (User's note: it may be better to use knots.zip below instead.)
  
• knots.zip: Java code for programs written by Michael An, Evan Liang, and Aninda Manocha, as part of the DRM@Duke program between the Duke mathematics department and NCSSM. The code updates Chongchitmate's original code and should be compiled before running. One new feature is a function that compares pairs of Legendrian knots given by grid diagrams and attempts to tell whether they are Legendrian isotopic.
  

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