Isotopies of Legendrian 1-knots and Legendrian 2-tori

TL;DR

This paper constructs Legendrian 2-tori from loops of Legendrian knots, explicitly computes their contact homology, and reveals new examples with unique properties, advancing understanding of Legendrian knotted tori.

Contribution

It introduces a method to construct Legendrian 2-tori from Legendrian knot loops and computes their contact homology explicitly, showing dependence on monodromy and producing novel examples.

Findings

Contact homology depends only on monodromy chain homotopy type.

Constructed Legendrian tori with non-augmentable DGA but non-trivial homology.

Found pairs of tori with identical linearized homology but different full contact homology.

Abstract

We construct a Legendrian 2-torus in the 1-jet space of $S^1\times\R$ (or of $\R^2$) from a loop of Legendrian knots in the 1-jet space of $\R$. 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.