Legendrian Mirrors and Legendrian Isotopy

We resolve a question of Fuchs and Tabachnikov by showing that there is a Legendrian knot in standard contact R with zero Maslov number which is not Legendrian isotopic to its mirror. The proof uses the differential graded algebras of Chekanov. A Legendrian knot in standard contact R is a knot which is everywhere tangent to the two-plane distribution induced by the contact one-form dz− y dx. Two Legendrian knots are Legendrian isotopic if there is a smooth isotopy between them through Legendrian knots. Fuchs and Tabachnikov [FT] define an involution of Legendrian knots as follows: given a Legendrian knot, let its Legendrian mirror be the image of the knot under the diffeomorphism (x, y, z) 7→ (x,−y,−z). (This terminology is due to [Che]; note, however, that our standard contact form has the opposite orientation.) Clearly a Legendrian knot and its mirror are isotopic as smooth knots; however, this isotopy is not through contactomorphisms. Thus there is no reason a priori why a knot should be Legendrian isotopic to its mirror. Nevertheless, an analysis of small examples shows that many knots are; Fuchs and Tabachnikov ([FT], repeated in [Tab] and [Che]) ask whether this is true in general. (They restrict this question to knots with zero Maslov number, since mirroring negates Maslov number; however, reversing knot orientation also negates Maslov number, and so one could ask more generally if an oriented knot is always Legendrian isotopic to its mirror, with the opposite orientation if necessary.) We will show that the answer is negative by displaying a counterexample. The proof uses the powerful differential graded algebra invariant defined by Chekanov [Che]. Let K be the (unoriented) Legendrian knot whose projection to the xy plane is given in Figure 1. (The diagram represents a Legendrian knot because, e.g., it is the morsification of a front diagram; see [Fer].) This knot has smooth isotopy type 62, Maslov index 0, and Thurston-Bennequin invariant −7. The Legendrian mirror M(K) of K is simply the reflection of this diagram about the x (horizontal) axis. Proposition The Legendrian mirrors K and M(K) are not Legendrian isotopic. Proof. From [Che], it suffices to show that the Chekanov DGAs corresponding to K and M(K) have nonisomorphic graded homology algebras. The Chekanov DGA Date: First version: 20 April 2000; this version: 28 August 2000. 2000 Mathematics Subject Classification. Primary 53D12; Secondary 57M27, 57R17.