On a remarkable polyhedron geometrizing the figure eight knot cone manifold

Abstract

The authors define a one-parameter family of polyhedra P(a), 0<a≤5−25√−−−−−−−√, in three-dimensional spaces of constant curvature −∞<k(a)≤1. Identifying faces of P(a) in pairs by isometries gives rise to cone manifolds M(a). For example, k=−1 when a=13−−√, and M gives the hyperbolic structure on the complement in S3 of the figure-eight knot K, k=0 when a=12−−√, and M gives the Euclidean structure on the orbifold which results from (3,0)-surgery on K, while k=1 when a=5−25√−−−−−−−√, and M gives the spherical structure on the orbifold which results from (2,0)-surgery on K. M(0) is a degenerate hyperbolic structure on the torus bundle B over S1 which results from (0,1)-surgery on K (let Σ⊂B denote the core circle of the surgery). The other M(a) interpolate between these, and after rescaling, as a increases, give hyperbolic structures on B, singular along Σ, with cone angles ranging from 2π to zero, then hyperbolic [resp. spherical] structures on S3, singular along K, with cone angles ranging from zero to 2π/3 [resp. 2π/3 to π]. Elementary formulas are derived for the volumes of the (rescaled) cone manifolds, the lengths of the (rescaled) singular sets, and the cone angles, as functions of a. Also, the phenomenon of "spontaneous surgery'' at a=13−−√ is linked to a combinatorial change in P(a).