Let M be a metric space such that the circle is deformation retract of M. If a circle is embedded in 3-space, we obtain a knot. Suppose the space M is embedded in 3-space and its projection image in a plane has n crossing points. Then is it true that there exists a knot K with n crossings such that K is deformation retract of the embedded M in 3-space.
I am thinking of the following case: Let $S_1^1$ and $S_2^1$ be two disjoint circles and $l$ is an arc joining the two circles. Pick a point $x_2 \in S_2^1$ and define a topological space $M=S_1^1 \cup \{ S_2^1 \setminus x_2 \} \cup l $. The subspace $S_1^1$ is deformation retract of M. Suppose M is embedded in 3-space by only knotted the arc $l$. The circle $S_1^1$ and the punctured circle $S_2^1 \setminus x_2$ are kept standard. Is the embedded M equivalent to some knot or there exists a knot K such that K is deformation retract of the embedded M.