Tag - Legendrian contact homology

Lenhard Ng: New Algebraic Invariants of Legendrian Links

24th May, 2024

For the past 25 years, Legendrian contact homology has played a key role in contact topology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids.

Austin Christian: Persistent Legendrian Contact Homology

12th April, 2024

This talk will report on an REU whose goal was to introduce the notion of persistence into Legendrian contact homology. The LCH of a Legendrian knot is computed as the homology of the knot's Chekanov-Eliashberg DGA and is a well-studied invariant of Legendrian isotopy types. For a given Legendrian embedding, the Chekanov-Eliashberg DGA admits a natural filtration, allowing for the computation of persistent homology. The purpose of this REU was to initiate the study of the resulting filtered homology.

Lenhard Ng: New Algebraic Invariants of Legendrian Links

25th March, 2024

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids.

Joshua Sabloff: Relative Calabi-Yau Structures for Legendrian Contact Homology

11th March, 2024

Legendrian Contact Homology (LCH) was among the first, and is still among the most important, non-classical invariants of Legendrian knots. In this talk, I will tell a story that builds up ever more sophisticated analogues of Poincare Duality in LCH. Despite the algebraic nature of the talk, I promise pictures and examples.

Soham Chanda: Augmentation Varieties and Disk Potential

26th January, 2024

Dimitroglou-Rizell-Golovko constructs a family of Legendrians in prequantization bundles by taking lifts of monotone Lagrangians. These lifted Legendrians have a Morse-Bott family of Reeb chords. We construct a version of Legendrian Contact Homology (LCH) for Rizell-Golovko's lifted Legendrians by counting treed disks. Our formalism of LCH allows us to obtain augmentations from certain non-exact fillings. We prove a conjecture of Rizell-Golovko relating the augmentation variety assoiciated to the LCH of a lifted Legendrian and the disk potential of the base Lagrangian. As an application, we show that lifts of monotone Lagrangian tori in projective spaces with different disk-potentials, e.g. as constructed by Vianna, produce non-isotopic Legendrian tori in contact spheres.