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Menasco, applying Thurston's hyperbolization theorem for Haken manifolds, showed that any prime, non-split alternating link is hyperbolic, i.e. the link complement has a hyperbolic geometry, unless the link is a torus link.
Poincaré disk model, a model of n-dimensional hyperbolic geometry
In 1910, following a 1909 publication of Sommerfeld, he applied hyperbolic geometry to the special theory of relativity.