An Atlas of Legendrian 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 seven crossings and nonalternating knots through nine 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 nonmaximal nondestabilizable Legendrian knots, and gives rise to new infinite families of transversely nonsimple knots.

2000 AMS Subject Classification:

• 57M25
• 57R17
• 53D42

Keywords:

• Legendrian knot
• transverse knot

ACKNOWLEDGMENTS

The authors would like to thank John Etnyre, Hiroshi Matsuda, Dan Rutherford, Josh Sabloff, and Shea Vela-Vick for illuminating discussions. Much of this work appeared in the first author's undergraduate honors thesis at Duke University, with support from the PRUV program at Duke. The second author was supported by NSF grant DMS-0706777 and NSF CAREER grant DMS-0846346.

Notes

¹Our online Legendrian knot and link atlas is available at http://alum.mit.edu/www/ng/atlas/ .

²The techniques of [Birman and Menasco 06, Birman and Menasco 08] are another approach to transverse nonsimplicity, but the knots of braid index 3 that they have proven to be transversely nonsimple all have arc index at least 10 and are not covered in the atlas.

³A note on conventions: to obtain grid diagrams as in Figure 3 for which we can apply as in, we either rotate a usual X–O diagram 90° counterclockwise and interchange X's and O's (for the third diagram in Figure 3), or rotate a usual X–O diagram 90° clockwise (for the first two diagrams). In the resulting diagrams (Figures 3 and 4), we use the convention from that horizontal segments pass over vertical segments. The resulting Legendrian front is either identical to the original front (for the third diagram) or related to the original front by the transformation L↦−μ(L) (for the first two diagrams; for both, the atlas states that this transformation is a Legendrian isotopy), possibly along with a few elementary moves in Gridlink (available at http://www.math.uic.edu/culler/gridlink/) .

⁴The Mathematica code for computing Legendrian invariants is available at http://www.haverford.edu/math/jsabloff/Josh_Sabloff/Research.html .