2020/02/28 by Jessica S. Purcell, Purcell, Jessica S. · 3 citations
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #30F40 #57M27 #57M50 #57Q15 #Algebraic geometry #Connective tissue disorders research #Engineering #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry #Hyperbolic geometry #Hyperbolic triangle #Knot (papermaking) #Knot invariant #Knot theory #Mathematics #Polyhedron #Primary: 57M25 #Pure mathematics #Secondary: 57N10 #Tricolorability #Twist #math.GT #msc:30F40 #msc:57M25 #msc:57M27 #msc:57M50 #msc:57N10 #msc:57Q15
paper · pdf · doi:10.48550/arxiv.2002.12652
published in arXiv (Cornell University) (Cornell University) · 344 pages, 158 figures
arxiv created 2020/02/28 · openalex publication_date 2020/02/28 · arxiv updated 2020/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric techniques used to study these families, such as angle structures and normal surfaces. The third part gives more detail on three important knot invariants that come directly from hyperbolic geometry, namely volume, canonical polyhedra, and the A-polynomial.