Tag - Legendrian knots

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.

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.

Bulent Tosun: Contact Surgeries and Symplectic Fillability

9th October, 2023

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, we will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (n) surgery along a Legendrian knot to yield a weakly fillable contact manifold, for some integer n > 0. When specialized to knots in the three sphere with its standard tight structure, this result can be effectively used to find many examples of fillable surgeries along with various obstructions and surprising topological applications. For example, we prove that a knot admitting lens space surgery must have slice genus equal to its 4-dimensional clasp number.

Angela Wu: On Lagrangian Quasi-Cobordisms

24th April, 2023

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a preorder, and that they are not symmetric, it is not known if they form a partial order on Legendrian knots. The idea of a Lagrangian quasi-cobordism was first defined by Sabloff, Vela-Vick, and Wong. Roughly, for two Legendrians of the same rotation number, it is the smooth composition of a sequence of alternatingly ascending and descending Lagrangian cobordisms which start at one knot and ends at the other. This forms a metric monoid on Legendrian knots, with distance given by the minimal genus between any two Legendrian knots. In this talk, I will discuss some new results about Lagrangian quasi-cobordisms, based on some work in progress with Sabloff, Vela-Vick, and Wong.

Georgios Dimitroglou Rizell: A Relative Calabi-Yau Structure for Legendrian Contact Homology

31st March, 2023

The duality long exact sequence relates linearised Legendrian contact homology and cohomology and was originally constructed by Sabloff in the case of Legendrian knots. We show how the duality long exact sequence can be generalised to a relative Calabi-Yau structure, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficiens in the Chekanov-Eliashberg DGA. This is joint work in progress with Legout.

Roger Casals: A Microlocal Invitation to Lagrangian Fillings

11th November, 2022

We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding Legendrian knots in the standard contact 3-sphere. Thanks to results from the microlocal theory of sheaves, which we will survey, we then show that this geometric problem gives rise to an interesting moduli space. In fact, we establish a bridge translating geometric operations, such as Lagrangian disk surgeries, into algebraic properties of this moduli space, such as the existence of cluster algebra structures. The talk is intended for a broad symplectic audience and all key ideas will be introduced and motivated.

Joshua Sabloff: Non-Orientable Lagrangian Fillings of Legendrian Knots

19th April, 2022

Most work on Lagrangian fillings of Legendrian knots to date has concentrated on orientable fillings, but instead I will present some first steps in constructions of and (especially) obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings.

Agniva Roy: Constructions of High-Dimensional Legendrians and Isotopies

25th March, 2022

I will talk about an ongoing project that explores the construction of high-dimensional Legendrian spheres from supporting open books and contact structures. The input is a Lagrangian disk filling of a Legendrian knot in the binding. We try to understand the relationship between different constructions from the same input, and suggest parallels, in the S2n+1 case, to a construction defined by Ekholm for ℝ2n+1.

Lisa Traynor: Legendrian Torus and Cable Links

22nd November, 2021

Legendrian torus knots were classified by Etnyre and Honda. I will explain the classification of Legendrian torus links. In particular, I will describe restrictions on the Legendrian torus knots that can be realized as the components of a Legendrian torus link, and I will give examples of Legendrian torus links that cannot be destabilized even though they do not have maximal Thurston-Bennequin invariant. Furthermore, I will explain that there are some smooth symmetries of Legendrian torus links that cannot be realized by a Legendrian isotopy. I will also describe how these torus link statements have extensions to Legendrian cable links. These results are applications of convex surface theory.