I have a couple of questions about a statement on this wiki page. It claims that the complement of a Hopf Link is a 'Hopf Band'. OK I can kind of see that if you imagine deforming their picture of a Hopf Band into a torus with a twisted hole in the center. Questions:
1) The page says that the Hopf band is 'one' Hopf Link complement. I thought link complements were unique?
2) I'm interested in the knot complement of the knot sum of two Hopf Links. Is there a way to compute that from two knot complements? I.e., is there a notion of a complement knot sum?
1) I think the caption writer confused "complement" for "Seifert surface."
2) There are two ways I know of for thinking about what knot sum means for the $3$-manifold.
(a) In $S^3$ itself, a knot sum is the existence of an embedded $S^2$ that intersects the knot in exactly two points. In the knot complement, this corresponds to a properly embedded annulus ("proper" meaning the boundary of the annulus intersects the boundary of the knot complement transversely) where the boundary curves are meridian curves.
In such a case, the boundary of the annulus cuts the tubular neighborhood of the knot into two pieces. Take one of these pieces and glue it to the annulus, and you get a torus, leading to JSJ decomposition:
(b) In the knot complement, the existence of an embedded incompressible torus. (This is more general than knot sums.)
So, a "complement knot sum" could be cutting out appropriate solid tori from each space and gluing the spaces together along the torus boundaries by the right homeomorphism.
Regarding Carl's point, in the above I'm using "knot sum" as a description of a decomposition --- it is not a well-defined operation for links of more than one component. Furthermore, for link complements you lose out on a well-defined meridian curve so you can't tell what are valid annuli to cut along if you want to do a sum of complements in (a)-style.
That's not to say you can't combine the Hopf links in some way, but you're supposed to specify how you are going to combine them. (It does matter in general for the fundamental group of a link complement: depending on which components of a link are the ones knot summed, the exact meridians that are amalgamated in the van Kampen theorem changes.)
That said, the Hopf links have link complement fundamental group $\mathbb{Z}\times\mathbb{Z}$, and every knot sum gives the same link, whose complement has the fundamental group with presentation $\langle a,t,b|[a,t]=1,[b,t]=1\rangle$.