Let $J,L$ be the two simple closed curves in the solid torus $T$ as shown below.
I am trying to show that $J$ is not null-homotopic in the complement of $L$. The hint given was to consider the universal cover of $T$ which would be an infinite cylinder $\mathbb{R} \times D^2$. From here, I don't know what to do. I presume we use the homotopy lifting property to restate this problem in terms of the preimages of $J$ and $L$ although I'm not sure. Any advice on how to proceed would be greatly appreciated
The method described in the hint works (start by drawing a picture of the universal cover and the chain of lifts of L and J, then look at two adjacent components and show they are not homotopically unlinked. If there are lifts that are homotopically linked, you can use the lifting property to show the originals must be homotopically linked). But it also seems easy enough to find generators for the fundamental group of $T-J$ and write $L$ as a word in the fundamental group. If it's a free group, then any nonreduced word (and in particular a word of the form $[a,b]$) is a nontrivial fundamental group element.