Isotopies of Legendrian 1-knots and Legendrian 2-tori

Abstract

We construct a Legendrian 2-torus in the 1-jet space of $S^1 x $\Bbb R$ (or of $\Bbb R^2$) from a loop of Legendrian knots in the 1-jet space of $\Bbb R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.