A survey of hyperbolic knot theory
David Futer, Efstratia Kalfagianni, and Jessica S. Purcell
TL;DR
This survey reviews tools and techniques for analyzing the geometric properties of hyperbolic links from diagrams, focusing on hyperbolicity, volume estimation, and cusp shape bounds, with applications and open questions.
Contribution
It compiles and discusses various methods for estimating geometric invariants of hyperbolic links directly from diagrams, highlighting recent progress and open problems.
Findings
Methods for determining hyperbolicity from diagrams
Techniques for estimating volume and cusp shapes
Open questions on geometric invariants
Abstract
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.
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