The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130:
He says that this dark-black knot in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \{ 0\} \subset S^1 \times D^2$, i.e., null-homologous.
How can we prove such a claim? Can we have a right to use tools of knots in $S^3$?