The smallest Haken hyperbolic polyhedra

Abstract

Abstract. We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.

Small volume link orbifolds

“…This is proved in Section 2; see Theorem 2.1 and Proposition 2.8 for precise statements. To prove the simple-seeming statement of Theorem 2.1, we draw on a number of results and techniques, including the orbifold geometrization theorem [8,11], Dunbar's classification of Euclidean orbifolds [12], collar estimates due to Gehring, Marshall, and Martin [17,18], and volume estimates for totally geodesic boundary due to Atkinson and Rafalski [7].…”

“…For each such triple (p, q, r), we construct and evaluate the volume estimate of Atkinson and Rafalski [7,Theorem 3.4], which is based on Miyamoto's theorem [25]. According to [7, We may also complete the proof of Theorem 2.1.…”