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.)


The canonical reference for this sort of thing is:

Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P., Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs. 5. Tokyo: Mathematical Society of Japan (MSJ). ix, 170 p. (2000). ZBL0955.57014.

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


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