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Is there any clear algorithm to compute the fundamental group of complements of knots?

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What do you mean by "compute"? There is the Wirtinger presentation ( but it is unclear what information you can extract from it in general. – Qiaochu Yuan Aug 12 '12 at 2:57
There's also an algorithm to construct a triangulation of the complement, from which there's a standard $\pi_1$ presentation construction. – Ryan Budney Aug 12 '12 at 5:39
the Wirtinger presentation was what I was looking for... Thanks guys – Sam Aug 13 '12 at 20:01

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