REU 2022: Legendrian contact homology
Mentors: Austin Christian and Agniva Roy
Participants: Edo Biluar, Sarah Brown, Shelly Cao, Michael Edwards, and Manyi Guo
The goal of this REU is was to study a modern algebraic invariant of Legendrian knots. A Legendrian knot is a knot which has a certain compatibility with the contact structure on \(\mathbb{R}^3\). Natural questions to ask about Legendrian knots include:
- How do we tell two Legendrian knots apart?
- Can we realize our favorite Legendrian knot as the boundary of a surface with nice geometry?
- Given such a surface, can we build surfaces for other knots?
The scariest words in the above description will be defined in the early stages of the REU --- you're not expected to have any familiarity with contact geometry. An undergraduate course in modern algebra (groups, rings, and such) and a recollection of 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
Background (which we'll cover in the REU)
- Adams, The Knot Book (PDF)
- Alsätra, Knots, Reidemeister Moves, and Knot Invariants (PDF)
- Armstrong, Basic Topology
- Bachman, A Geometric Approach to Differential Forms (arXiv link)
- Rolfsen, Knots and Links (PDF)
Survey papers
- Etnyre, Legendrian and Transversal Knots (arXiv link)
- Etnyre & Ng, Legendrian contact homology in \(\mathbb{R}^3\) (arXiv link)
Research papers
- Capovilla--Searle, Infinitely many planar fillings and symplectic Milnor fibers (arXiv link)
- Capovilla--Searle-Traynor, Non-Orientable Lagrangian Cobordisms between Legendrian Knots (arXiv link)
- Casals-Ng, Braid Loops with infinite monodromy on the Legendrian contact DGA (arXiv link)
- Cornwell-Ng-Sivek, Obstructions to Lagrangian concordance (arXiv link)
- Ekholm-Honda-Kálmán, Legendrian knots and exact Lagrangian cobordisms (arXiv link)
- Pan, The augmentation category map induced by exact Lagrangian cobordisms (arXiv link)
- Pan-Rutherford, Augmentations and immersed Lagrangian fillings (arXiv link)
- Wu, Obstructing Lagrangian concordance for closures of 3-braids (arXiv link)
Notes
Here are some notes written by Agniva and me for our background sections.
- Legendrian Knots I (by Agniva)
- Austin's week 1 notes
- Legendrian Knots II (by Agniva)
- DGAs II (by Agniva)
- Invariance, front projections, and links (by Austin)
- Observations about the Chekanov-Eliashberg DGA (by Agniva)
- Computable invariants (by Austin)
- Cobordisms, fillings, and augmentations (by Agniva)
- Some nonsense about Lagrangian cobordisms (by Austin)
And here are the exercises for our problem sessions: PDF
Final reports
Product structure on pairs in Legendrian knot atlas, Shelly Cao, Michael Edwards, Manyi Guo
Legendrian connected sums, Edo Biluar, Sarah Brown