I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help.
Suppose $K$ and $L$ are two curves inside the solid torus $\mathbb S^1 \times D$, which are each on one side of the hole, and intersect each other like chain links at each end of the loop. Can we show that K is not null-homotopic in $(\mathbb S^1 \times D) - L$?
It is really clear to me that if we are thinking of these curves as existing in just $\mathbb R^3$ that we can unlink the two by passing one curve around the other - but that we can't do this in the torus because the central hole stops you from deforming the curves to be close together. I suspect something like this is key, but I don't know how to write down the proof. Any help would be greatly appreciated.
