Augmentations of Legendrian Knots

Description: Legendrian knot is one of the current interests in knot theory. The main idea Legendrian setting is to give the underlying space a contact structure which essentially equips the space with a plane field which can keep track of how knots turn within the space. There are a few classical invariants of Legendrian knots including rotation number and Thurston-Benniquin’s number. There were questions whether these two invariants are sufficient for distinguishing non Legendrian isotopic knots. To that end the new invariant called contact homology arose via the study of holomorphic curves. Recently, Chekanov gave a combinatorial description of contact homology and found a class of non Legendrian isotopic knots which cannot be distinguishes by classical invariants. In doing so, Chkanov associated a so call augmentation maps to the contact homology. The set of all augmentations associated to a Legendrian knot seems to contain all the information included in the corresponding contact homology. I am currently studying the behavior of the augmentations with respect to the change in Legendrian isotopy type.

Supap Kirtsaeng3

University of Southern California