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,
$ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$
and
$ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$
have the same homotopy
$(2n-7)$
-type. In the four-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
$\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$
. We use our model to show that for every choice of basepoint, each of the homotopy groups,
$\pi _1$
and
$\pi _2,$
of
$ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$
contains an infinitely generated free abelian group.