  Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension $n$
that are homeomorphic. We prove that the spaces of smooth knots $Emb(S^1, N_1)$
and $Emb(S^1, N_2)$ have the same homotopy $(2n-7)$-type. In the 4-dimensional
case this means that the spaces of smooth knots in homeomorphic 4-manifolds
have sets $\pi_0$ of components that are in bijection, and the corresponding
path components have the same fundamental groups $\pi_1$. The result about
$\pi_0$ is well-known and elementary, but the result about $\pi_1$ appears to
be new. The result gives a negative partial answer to a question of Oleg Viro.
Our proof uses the Goodwillie-Weiss embedding tower. We give a new model for
the quadratic stage of the Goodwillie-Weiss tower, and prove that the homotopy
type of the quadratic approximation of the space of knots in $N$ does not
depend on the smooth structure on $N$. Our results also give a lower bound on
$\pi_2 Emb(S^1, N)$. We use our model to show that for every choice of
basepoint, each of the homotopy groups $\pi_1$ and $\pi_2$ of $Emb(S^1,
S^1\times S^3)$ contains an infinitely generated free abelian group.
