If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of *embedding calculus*. We set $T_\infty F := \mathrm{holim} T_k F$.

The functor of most interest is the functor of embeddings $\mathrm{Emb}(-,N)$ into a fixed target $N$. By work of Goodwillie, Klein, and Weiss, $T_\infty\mathrm{Emb}(M,N)\simeq\mathrm{Emb}(M,N)$ as long as $\dim N-\dim M\geq3$. One says that "the Taylor tower converges."

I am interested in examples below codimension 3. One such example is knot theory, where it is known that $\mathrm{Emb}(S^1,\mathbb{R}^3)\not\simeq T_\infty\mathrm{Emb}(S^1,\mathbb{R}^3).$ This is the only example I know, and I would like to change this. Specifically, I am looking for examples of manifolds $M$ and $N$ such that either

1) $\dim N-\dim M<3$ and the Taylor tower is known

notto converge, or2) $\dim N-\dim M<3$ and the Taylor tower converges anyway.