Is there a simple proof of the fact that if $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$ embedded into $S^3\setminus A$ that such that the circles $A$ and $B$ are linked with the linking number $1$? If $A$ is smoothly embedded, than it is easy. The problem is that the general topological embedding can be very complicated and I do not see a simple geometric explanation of this fact. I know that $H_1(S^3\setminus A)=\mathbb Z$ (Corollary 1.29 in Vick's Homology Theory) so the generator of the homology group should be a circle.

This is actually true as John Klein pointed out in his answer below, but I do not know the answer to the higher dimensional version of the problem stated below. Only a partial answer is given in comments below.

The question can be then generalized to higher dimensional spheres. If $A\subset S^n$ is homeomorphic to $S^k$, there should be a topological sphere in $S^n\setminus A$ of dimension $n-k-1$ that links $A$ with the linking number $1$.

Is there a simple proof of the fact that:

If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$ embedded into $S^3\setminus A$ that such that the circles $A$ and $B$ are linked with the linking number $1$?

If $A$ is smoothly embedded, than it is easy. The problem is that the general topological embedding can be very complicated and I do not see a simple geometric explanation of this fact. I know that $H_1(S^3\setminus A)=\mathbb Z$ (Corollary 1.29 in Vick's Homology Theory) so the generator of the homology group should be a circle.

Edit 1: This is actually true as John Klein pointed out in his answer below, but I do not know the answer to the higher dimensional version of the problem stated below. Only a partial answer is given in comments below.

The question can be then generalized to higher dimensional spheres.

If $A\subset S^n$ is homeomorphic to $S^k$, there should be a topological sphere in $S^n\setminus A$ of dimension $n-k-1$ that links $A$ with the linking number $1$.

Edit 2. A counterexample is given below in my answer.

Is there a simple proof of the fact that if $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$ embedded into $S^3\setminus A$ that such that the circles $A$ and $B$ are linked with the linking number $1$? If $A$ is smoothly embedded, than it is easy. The problem is that the general topological embedding can be very complicated and I do not see a simple geometric explanation of this fact. I know that $H_1(S^3\setminus A)=\mathbb Z$ (Corollary 1.29 in Vick's Homology Theory) so the generator of the homology group should be a circle.

The question can be then generalized to higher dimensional spheres. If $A\subset S^n$ is homeomorphic to $S^k$, there should be a topological sphere in $S^n\setminus A$ of dimension $n-k-1$ that links $A$ with the linking number $1$.

Is there a simple proof of the fact that if $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$ embedded into $S^3\setminus A$ that such that the circles $A$ and $B$ are linked with the linking number $1$? If $A$ is smoothly embedded, than it is easy. The problem is that the general topological embedding can be very complicated and I do not see a simple geometric explanation of this fact. I know that $H_1(S^3\setminus A)=\mathbb Z$ (Corollary 1.29 in Vick's Homology Theory) so the generator of the homology group should be a circle.

This is actually true as John Klein pointed out in his answer below, but I do not know the answer to the higher dimensional version of the problem stated below. Only a partial answer is given in comments below.

The question can be then generalized to higher dimensional spheres. If $A\subset S^n$ is homeomorphic to $S^k$, there should be a topological sphere in $S^n\setminus A$ of dimension $n-k-1$ that links $A$ with the linking number $1$.