Let K₀ and K be knots in R³. Suppose that by a compactly supported Hamiltonian isotopy on T^*R³, the conormal bundle of K₀ 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 K₀ and K. One example is that if K₀ is the unknot, then K is also the unknot. We also consider some cases where K₀ 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 K₀ and K determined by these DGAs.
Yukihiro Okamoto (Thu,) studied this question.