On knots in the zero section which appear as clean Lagrangian intersections

Abstract

Let \(K_0\) and \(K\) be knots in \(\mathbb{R}^3\). Suppose that by a Hamiltonian isotopy on \(T^*\mathbb{R}^3\) with a compact support, the conormal bundle of \(K_0\) is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along \(K\). Then, we prove that \(K_0\) and \(K\) have a relation on the framed knot DGA, which is a knot invariant defined by Ng. One of its consequences is that if \(K_0\) is the unknot, then \(K\) is also the unknot. These results are derived from studies on the Chekanov-Eliashberg DGAs and the DGA maps induced by Lagrangian cobordisms.