2008/04/30 by Ian Agol · 5 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:30F40 #msc:57M
paper · pdf · doi:10.1090/s0002-9939-10-10364-5
published as electronically published May 12, 2010, Proc. Amer. Math. Soc. (2010), http://www.ams.org/journals/proc/0000-000-00/S0002-9939-10-10364-5/home.html · 10 pages, 5 figures, incorporated referees comments, includes hyperref links
openalex publication_date 2010/05/12 · arxiv created 2010/05/18 · arxiv updated 2010/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove that the Whitehead link complement and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative 2 comma 3 comma 8 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(-2,3,8)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3.66 period period period"> <mml:semantics> <mml:mn>3.66...</mml:mn> <mml:annotation encoding="application/x-tex">3.66...</mml:annotation> </mml:semantics> </mml:math> </inline-formula> = 4 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="times"> <mml:semantics> <mml:mo> × </mml:mo> <mml:annotation encoding="application/x-tex">×</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Catalan’s constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.