On Archimedean link complements

Abstract

We study a subclass of alternating links for which the the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.