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Sep 22, 2013 at 7:23 comment added Neil Hoffman @IgorRivin You are right that it's not enough to construct the manifolds. However, often you can get an integral representation of manifold over or if you are using software to guess at exact values from approximate ones, the tables in MacLachlan-Reid are a good sanity check. Finally, I forgot to mention Riley's Parabolic representations of knot groups I, plms.oxfordjournals.org/content/s3-24/2/217.full.pdf might be useful for you. In particular, Theorem 2 addresses your concerns directly.
Sep 21, 2013 at 14:03 comment added Igor Rivin Thanks! I will check these out (re Maclachlan-Reid, presumably invariant trace fields are not enough to actually reconstruct the manifold, so hopefully there is more info..)
Sep 21, 2013 at 7:24 history answered Neil Hoffman CC BY-SA 3.0