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
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.
- May 2011: added new data about
transverse nonsimplicity derived from the transverse homology
- 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.
The Legendrian knot atlas
This is the main atlas, with explanation, as a standalone file. Previous version: August 2013 Legendrian knot atlas.
Atlas for Legendrian two-component
Legend/explanation for the Legendrian
two-component link atlas
Classification of Legendrian knots
and links by Wutichai Chongchitmate: senior thesis, Duke
- 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
input into the Mathematica program “Legendrian
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
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
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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