2016/10/25 by Autumn E. Kent, Kent, Autumn E., Jessica S. Purcell +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #30F40 #57M25 #57M50 #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #msc:30F40 #msc:57M25 #msc:57M50
paper · pdf · doi:10.48550/arxiv.1610.07731
V3: 10 pages, 1 figure. Minor changes. To appear in Mathematical Research Letters. V2: 10 pages, 1 figure. Details added to proof of lemma 4.1, as well as minor revisions elsewhere. V1: 8 pages, 1 figure
openalex publication_date 2016/10/25 · arxiv created 2018/06/21 · arxiv updated 2018/06/25 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at least R; we call this the R-thick part of the manifold. We show that for any ε>0, there exists a knot K in the 3-sphere so that the ratio of the volume of the R-thick part of the knot complement to the volume of the knot complement is at least 1-ε. As R approaches infinity, and as ε approaches zero, this gives a sequence of knots that is said to Benjamini--Schramm converge to hyperbolic space. This answers a question of Brock and Dunfield.