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Can someone give me a reference for a proof that the Dehn presentation of a knot group gives us the fundamental group of the knot complement in $S^{3}$?

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Please don't cross-post on MO. – user38268 Dec 27 '12 at 2:54

Are you referring to the Wirtinger presentation?

The Dehn complex is a complex with two vertices, one above the plane of the knot projection, and one below, and an edge corresponding to each complementary region of the knot projection, and square associated to each crossing. The knot complement deformation retracts to this complex, so the fundmental group of the knot complement is the fundamental group of the complex. See this article for an explicit description of the Dehn complex in a more general context. To get the Wirtinger presentation (or Dehn presentation?), one may take a basis of loops in the 2-skeleton corresponding to edges of the knot projection diagram, where one has a loop sewing through one adjacent face, and then threading back through the face on the other side of the edge. One then gets a relation associated to each crossing.

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