REU 2023: Persistent Legendrian contact homology
Mentors: Daniel Irvine and Austin Christian
TA: Weizhe Shen
Participants: Maya Basu, Ethan Clayton, and Fredrick Mooers
Update: This REU resulted in the paper Persistent Legendrian contact homology in \(\mathbb{R}^3\) .
In this REU, we will attempt to apply some techniques from topological data analysis to a problem in contact topology. Namely, we will use the action filtration on Legendrian knots in the standard contact manifold \((\mathbb{R}^3,\xi_0)\) to compute a persistent analogue of (linearized) Legendrian contact homology.
A Legendrian knot is a knot which has a certain compatibility with the standard contact structure on \(\mathbb{R}^3\), and Legendrian contact homology is a powerful modern invariant for distinguishing Legendrian knots up to isotopy (among other applications). Persistent homology, on the other hand, is a tool of topological data analysis which allows for the study of a given space at various resolutions. The underlying space which gives rise to Legendrian contact homology admits a natural filtration, and this filtration provides the various "resolutions" at which we will study our problem.
The early stages of the REU will be used to develop the requisite background --- you're not expected to have any familiarity with contact geometry or persistence modules. A very strong understanding of undergraduate modern algebra (groups, rings, fields, and such) and multivariable calculus should provide adequate academic background. The most important expectation of students is that they are interested in challenging themselves with an exciting new topic!
Reading list
Survey papers
- Etnyre, Legendrian and Transversal Knots (arXiv link)
- Etnyre & Ng, Legendrian contact homology in \(\mathbb{R}^3\) (arXiv link)
- Edelsbrunner & Harer, Persistent Homology -- a Survey (PDF)
Research papers
- Carlsson & Zomorodian, Computing persistent homology (Springer)
- Civan, Etnyre, Koprowski, Sabloff, & Walker, Product structures for Legendrian contact homology (arXiv link)
- Dimitroglou Rizell & Sullivan, The persistence of the Chekanov-Eliashberg algebra (arXiv link)
- Sabloff & Traynor, The Minimal Length of a Lagrangian Cobordism between Legendrians (arXiv link)
Notes
Here are some notes written by Daniel and Austin for our background sections.
- Legendrians in \(\mathbb{R}^3\) (Austin)
- Differential graded algebras (Daniel)
- The Chekanov-Eliashberg DGA (Austin)
- Persistence modules (Daniel)
- Properties of the Chekanov-Eliashberg DGA (Austin)
- Linearized Legendrian contact homology (Austin)
- Products on the Chekanov-Eliashberg DGA (Daniel)
- A Legendrian knot distinguished from its mirror by products on LCH (Austin)
And here are the exercises for our problem sessions: PDF.