We consider smooth embeddings of $S^1\sqcup S^2$ inside $S^4$. Let $f$ be such an embedding with the property that $f|_{S^2}$ is isotopic to the standard embedding, say in the first 3 coordinates. Let $\textrm{Emb}(S^1\sqcup S^2,S^4)_\textrm{std}$ denote the isotopy class of such embeddings as above. This is also an abelian group via embedded connect sum operation. [The notion of isotopy and ambient isotopy may not coincide as the codimension is $2$. Hence, the questions I am asking below are applicable for both notions, if they differ.]

(1) Is it true that $\textrm{Emb}(S^1\sqcup S^2,S^4)_\textrm{std}$ is infinite cyclic?

**Remark**: While reading Habegger's 1986 paper *Knots & links in codimension greater than 2* (available here), it seems that one can deduce (1) via Theorem 1.2 although this is contradicted when one uses Corollary 1.3. This may be due to some typo's in the formula or lack of my understanding.

(2) If we remove the assumption of the embedding being standard on $S^2$, then we are led to consider $\textrm{Emb}(S^1\sqcup S^2,S^4)$. How much of the group structure of $\textrm{Emb}(S^1\sqcup S^2,S^4)$ is known in terms of $\textrm{Emb}(S^2,S^4)$?

(3) More generally, is it known that $\textrm{Emb}(S^1\sqcup S^{n-2},S^n)_\textrm{std}$ is infinite cyclic? [It seems to be the case based on Habegger's paper but I am stuck at the same confusion/typo as pointed above.] How much of $\textrm{Emb}(S^1\sqcup S^{n-2},S^n)$ can be captured by $\textrm{Emb}(S^{n-2},S^n)$?

Edit: As pointed out by D. Ruberman, $\textrm{Emb}(S^1\sqcup S^{n-2},S^n)$ need not be a group. However, even for $n=2$, I'm interested in seeking a characterization of the embeddings in terms of $\textrm{Emb}(S^2,S^4)$ and the linking number of $S^1$ and $S^2$ in $S^4$.