Consider mutually transversal copies $\zeta_k=(\Bbb H^2 \times \Bbb R)_k$ each geodesically compactified and consider $M=\bigcup_{k=1}^4 \partial \zeta_k$. What knots are permissible on $M$? I note that $M$ is self-intersecting and immersed in $\Bbb R^3$. To at least get started we can write down an embedding $ e: S^1\longrightarrow M .$ The unknot exists on $M$ and I think the trefoil is exists as well. I feel like there is a restricted class of knots existing on $M$ but don't know how to identify this class precisely.
What knots exist on $M$?
