  Let $\mathrm{Emb}(S^1,M)$ be the space of smooth embeddings from the circle
to a closed manifold $M$ of dimension $\geq 4$. We study a cosimplicial model
of $\mathrm{Emb}(S^1,M)$ in stable categories, using a spectral version of
Poincar\'e-Lefschetz duality called Atiyah duality. We actually deal with a
notion of a comodule instead of the cosimplicial model, and prove a comodule
version of the duality. As an application, we introduce a new spectral sequence
converging to $H^*(\mathrm{Emb}(S^1,M))$ for simply connected $M$ and for major
coefficient rings. Using this, we compute $H^*(\mathrm{Emb}(S^1, S^k\times
S^l))$ in low degrees with some conditions on $k$, $l$. We also prove the
inclusion $\mathrm{Emb}(S^1,M)\to \mathrm{Imm}(S^1,M)$ to the immersions
induces an isomorphism on $\pi_1$ for some simply connected $4$-manifolds,
related to a question posed by Arone and Szymik. We also prove an equivalence
of singular cochain complex of $\mathrm{Emb}(S^1,M)$ and a homotopy colimit of
chain complexes of a Thom spectrum of a bundle over a comprehensible space. Our
key ingredient is a structured version of the duality due to R. Cohen.
