Relative Contact Homology - Legendrian Knots

Legendrian Knots

The simplest case yields invariants of Legendrian knots inside contact three-manifolds. The relative contact homology has been shown to be a strictly more powerful invariant than the "classical invariants", namely Thurston-Bennequin number and rotation number (within a class of smooth knots).

Yuri Chekanov developed a purely combinatorial version of relative contact homology for Legendrian knots, i.e. a combinatorially defined invariant that reproduces the results of relative contact homology.

Tamas Kalman developed a combinatorial invariant for loops of Legendrian knots, with which he detected differences between the fundamental groups of the space of smooth knots and of the space of Legendrian knots.