# Austin Christian

Georgia Tech Mathematics

### REU 2022: Legendrian contact homology

##### 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?
Some keywords for the above questions include Legendrian isotopy, Lagrangian filling, and Lagrangian concordance, but we'll learn those in due time. Our primary tool for addressing these questions will be Legendrian contact homology, which associates to each Legendrian knot a differential graded algebra. This algebra can be very complicated, but, even without a full understanding of the algebra, we can make interesting conclusions about Legendrian 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!

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.

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