You could consider images such as yours to be a knot in $(S^1)^3$ -- a single-component embedded 1-manifold in the "3-torus" $(S^1)^3$. In your case you get the 3-torus by modding out by your periodicity lattice. In your image there's a rank two part of the lattice in the horizontal plane, and one direction to the lattice that's in a not quite up-down direction, but a little skewed.

Knot theory in $(S^1)^3$ is very similar to knot theory in $S^3$. In fact, there's a way to cook up knots and links in $S^3$ from knots in $(S^1)^3$.

Case 1: you have a knot in $(S^1)^3$ and its homologically trivial. In this case you have a traditional alexander polynomial invariant. This isn't what's going on in your picture, though. The nice thing about this situation is that your knot lifts up to a link in the universal cover. So the components themselves have traditional knot invariants, and you also have various choices of lifts which give choices of links, and they all have invariants of their own.

Case 2: the knot $K$ is homologically non-trivial in $(S^1)^3$. This is what's happening in your picture. In this case, the $H_1$ of the complement of the knot maps onto a free group of rank 2. Moreover, the covering space corresponding to this cover is the complement of a knot in a solid torus. Precisely, the homology of the knot complement is isomorphic to the homology of $(S^1)^3$. $H_1$ of the complement maps onto $H_1((S^1)^3) / H_1(K)$, which is free of rank 2 plus a possible finite cyclic group. Take the map onto the free part.

The covering space corresponding to this map is diffeomorphic to $S^1 \times D^2$, which is an unknot complement. Moreover, since the knot's fundamental class maps to zero in this group, the knot lifts to a knot in this covering space. Putting this all together, associated to a knot in $(S^1)^3$ which is homologically non-trivial you get a $2$-component link in $S^3$ where one component is unknotted. Like case 1, you could make this a many-component link by taking various other lifts of $K$. So there's knot and link invariants associated to these objects, and they're invariants of your original periodic link. In your case, depending on which lifts you take you get components that link each other.

Is there anything more particular you'd like to know about these kinds of invariants? A group of condensed-matter physicists that study knotted light asked me about these kinds of things earlier this summer.