LEGEND FOR LEGENDRIAN KNOT ATLAS

The table depicts conjectural classifications of Legendrian knots in all prime knot types of arc index up to 9. • For each knot, the non-destabilizable Legendrian representatives are depicted (modulo the symmetries described below), with their (tb, r), along with the conjectural mountain range. As usual, rotate 45◦ counterclockwise to translate from grid diagrams to fronts. • Legendrian classification is known for torus knots and 41 [4], and for twist knots [5]. In the table, torus knots are denoted in the usual way by T (p, q), and the twist knot with n half-twists is denoted by Kn. • It is interesting to compare this atlas to the table from [6]. It would be interesting to know which of the Legendrian knots can be dis-tinguished via various modern techniques: Massey products, SFT, etc. • The (tb, r) coordinates for the mountain ranges can be inferred from the other (tb, r) information. Boxes enclose dots with the same (tb, r). Mountain ranges extend downward from the depicted parts in the usual way; mountain ranges without boxes indicate knot types that are conjecturally Legendrian simple. • The conjectured mountain ranges include black and red dots, one dot for each Legendrian isotopy type; the black dots represent a “lower-bound ” range of Legendrian types that we have been able to distinguish using current techniques, while the red dots are Legen-drian knots that we believe to be distinct from the black dots. We believe that the mountain ranges are complete as shown—i.e., that there are no other Legendrian knots besides the ones depicted—but this is only known for the knot types that have been classified. • Two symmetries on Legendrian knots are: orientation reversal, L 7→ −L, and Legendrian mirror, L 7 → µ(L) (reflecting fronts in the x axis). Both of these negate rotation number. Orientation reversal also changes topological type in general, but all of the knots in the atlas are invertible. Each Legendrian knot L thus produces up to four knots: L, −L, µ(L), −µ(L). The table depicts one representative from each of these orbits of up to four knots, along with information about which of the four knots in the orbit are isotopic, if any. For knots with nonzero rotation number, we choose a representative L with positiv