  Let $K_0$ and $K$ be knots in $\mathbb{R}^3$. Suppose that by a compactly
supported Hamiltonian isotopy on $T^*\mathbb{R}^3$, 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.
