$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, the second lies in the $y=0$ plane and its center is the point $(1,0,0)$)

Is it true that the circles $A$ and $B$ cannot be separated by a set that is homeomorphic to the $2$-sphere? (The homeomorphism can be arbitrarily “bad”, as in the case of the horned Alexander sphere)

(A set $S$ separates $A$ and $B$ iff $A$ and $B$ are subsets of different сonnected components of $\mathbb{R}^3\setminus S$)

I know how to solve this problem in a "smooth" case using knot theory. But this solution doesn't work when the embedding of the $2$-sphere is arbitrarily "bad".

$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, the second lies in the $y=0$ plane and its center is the point $(1,0,0)$.)

Is it true that the circles $A$ and $B$ cannot be separated by a set that is homeomorphic to the $2$-sphere? (The homeomorphism can be arbitrarily “bad”, as in the case of the horned Alexander sphere.)

(A set $S$ separates $A$ and $B$ iff $A$ and $B$ are subsets of different сonnected components of $\mathbb{R}^3\setminus S$.)

I know how to solve this problem in a "smooth" case using knot theory. But this solution doesn't work when the embedding of the $2$-sphere is arbitrarily "bad".

A and B are two linked (geometric) circles in $\mathbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, the second lies in the $y=0$ plane and its center is the point $(1,0,0)$)

Is it true that the circles A and B cannot be separated by a set that is homeomorphic to the 2-sphere? (The homeomorphism can be arbitrarily “bad”, as in the case of the horned Alexander sphere)

(A set S separates A and B iff A and B are subsets of different сonnected components of $\mathbb{R}^3\setminus S$)

I know how to solve this problem in a "smooth" case using knot theory. But this solution doesn't work when the embedding of the 2-sphere is arbitrarily "bad"

$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, the second lies in the $y=0$ plane and its center is the point $(1,0,0)$)

Is it true that the circles $A$ and $B$ cannot be separated by a set that is homeomorphic to the $2$-sphere? (The homeomorphism can be arbitrarily “bad”, as in the case of the horned Alexander sphere)

(A set $S$ separates $A$ and $B$ iff $A$ and $B$ are subsets of different сonnected components of $\mathbb{R}^3\setminus S$)

I know how to solve this problem in a "smooth" case using knot theory. But this solution doesn't work when the embedding of the $2$-sphere is arbitrarily "bad".