A hyperbolic link is one whose complement admits a hyperbolic metric. 
Hyperbolic links, and especially hyperbolic knots, are quite popular these days. However, I am currently interested in links whose complement admits a flat (i.e. locally euclidean) metric. If I got it right, the major difference from the hyperbolic case is the existence 
(at least, I want for it to exist) of a natural compactification. This compactification makes the 3-sphere into an Alexandrov space with conical singularities along the components of the link, or something like this. Is there such a construction somewhere in the literature?


[EDIT] I think I need to clarify this a bit. What I want is a metric on ${\mathbb S}^3$ such that 


1) The metric is flat outside of the link.


2) Each component of the link has a neighborhood isometric to a product of a conical point by ${\mathbb S}^1$. 


(So, if it is a ``negative'' conical point, then it won't be an Alexandrov space.)



"Flat links", a reference request
The canonical reference for this sort of thing is:


*Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P.*, [**Three-dimensional orbifolds and cone-manifolds**](http://dx.doi.org/10.2969/msjmemoirs/005010000), MSJ Memoirs. 5. Tokyo: Mathematical Society of Japan (MSJ). ix, 170 p. (2000). [ZBL0955.57014](https://zbmath.org/?q=an:0955.57014).


(available for free on Daryl Cooper's web page at UCSB).


