Tag - Legendrian knots

This talk is based on a joint work with Thomas Kragh. Using the generating function theory we split inject homotopy groups of pseudo-isotopy and/or h-cobordism spaces into various spaces of Legendrian manifolds, e.g. the space of Legendrian unknots in ℝ²ⁿ⁺¹ for a sufficiently large n. For instance, there is a non-trivial element in π₂ of the space of Legendrian unknots in ℝ²ⁿ⁺¹ for n ≥ 12.

Given a smooth knot K in the 3-sphere, a classic question in knot theory is: What surfaces in the 4-ball have boundary equal to K? One can also consider immersed surfaces and ask a 'geography' question: What combinations of genus and double points can be realized by surfaces with boundary equal to K? I will discuss symplectic analogues of these questions: Given a Legendrian knot, what Lagrangian surfaces can it bound? What immersed Lagrangian surfaces can it bound? These Lagrangian surfaces are commonly called Lagrangian fillings of the Legendrian knot and are more rigid than their topological counterpart. In particular, while any smooth knot bounds an infinite number of topologically distinct surfaces, there are classical and non-classical obstructions to the existence of Lagrangian fillings of Legendrian knots. Specifically, a polynomial associated to the Legendrian boundary through the technique of generating families can show that there is no compatible embedded Lagrangian filling. Immersed Lagrangian fillings are more flexible, and I will describe how this polynomial associated to the Legendrian boundary forbids particular combinations of genus and double points in immersed Lagrangian fillings. In addition, I will describe some constructions of immersed fillings that allow us to completely answer the Lagrangian geography question for some Legendrian knots.

We present a full h-principle (relative, parametric, C⁰-close) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy-theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The main ingredients in the proof are a refinement of the holonomic approximation lemma and the construction of a local wrinkling model for Lagrangian and Legendrian submanifolds. We give sample applications of our h-principle, including an Igusa-type theorem which states that higher singularities are unnecessary for the homotopy-theoretic study of the space of Legendrian knots in the standard contact Euclidean 3-space. This last result can be understood as a generalization of the Reidemeister theorem for families of Legendrian knots parametrized by a space of arbitrarily high dimension.