Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. Nadler-Zaslow correspondence suggests a connection between the two types of invariants. This dissertation presents a sequence of results exploring this connection. First, we define a Radon transform for sheaf categories. We prove the transform is an equivalence after a proper localization on each side. As an application, we show that two sheaf categories, both of which are knot invariants, are equivalent up to local systems. Second, we give a classification of abelian sheaves in Euclidean three space simple along the Legendrian knot conormal. The classification addresses the underlining reason why KCH representations lead to specialized augmentations. We also give sheaf-theoretical interpretations to related invariants such as the two-variable augmentation polynomial. Finally, we use Legendrian probes to compute stalks of the sheaf associated to an augmentation. In the case of the unknot conormal, the probing calculation shows a bifurcation of sheaf stalks and we also find the non trivial homotopy between the stalks.

Last modified

• 03/29/2018

Creator

• Honghao Gao

DOI

• https://doi.org/10.21985/N21105

Subject

• Mathematics

Keyword

• knot invariant
• symplectic topology
• contact topology
• microlocal sheaf theory
• knot contact homology

Date created

• 2017-01-01

Resource type

• Dissertation

Rights statement

• In Copyright