## Journal of Symplectic Geometry

### 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.

#### Article information

Source
J. Symplectic Geom., Volume 6, Number 4 (2008), 407-460.

Dates
First available in Project Euclid: 15 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1232029298

Mathematical Reviews number (MathSciNet)
MR2471099

Zentralblatt MATH identifier
1206.57030

#### Citation

Ekholm, Tobias; Kálmán, Tamás. Isotopies of Legendrian 1-knots and Legendrian 2-tori. J. Symplectic Geom. 6 (2008), no. 4, 407--460. https://projecteuclid.org/euclid.jsg/1232029298