J-braid groups are torus necklace groups

We construct a family of links we call torus necklaces for which the link groups are precisely the braid groups of generalised $J$-reflection groups. Moreover, this correspondence exhibits the meridians of the aforementioned link groups as braid reflections. In particular, this construction generalises to all irreducible rank two complex reflection groups a well-known correspondence between some rank two complex braid groups and some torus knot groups. In addition, as abstract groups, we show that the family of link groups associated to Seifert links coincides with the family of circular groups. This shows that every time a link group has a non-trivial center, it is a Garside group.