On knot types of clean Lagrangian intersections in $T^{\ast }{\mathbb{R}}^3$

Abstract

Let $K_0$ and $K$ be knots in ${\mathbb{R}}^3$. Suppose that by a compactly supported Hamiltonian isotopy on $T^{\ast }{\mathbb{R}}^3$, the conormal bundle of $K_0$ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along $K$. In this paper, we prove some constraints on the pair of knot types of $K_0$ and $K$. One example is that if $K_0$ is the unknot, then $K$ is also the unknot. We also consider some cases where $K_0$ and $K$ have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of $K_0$ and $K$ determined by these DGAs.

arXiv comment47 pages, minor revision following referee's comments, accepted for publication in Journal of Symplectic Geometry