This book provides an introduction to hyperbolic geometry in dimension three, with
motivation and applications arising from knot theory. Hyperbolic geometry was first used as …

" Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics.
This enyclopedia is filled with valuable information on a rich and fascinating subject."–Ed …

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above
by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot …

We present explicit geometric decompositions of the hyperbolic complements of alternating
k-uniform tiling links, which are alternating links whose projection graphs are k-uniform …

We study collections of planar curves that yield diagrams for all knots. In particular, we show
that a very special class called potholder curves carries all knots. This has implications for …

We present two methods to decompose a link complement into bipyramids, which have a
variety of applications. Here, we apply them to obtain new upper bounds on volume for …

The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume
of K to the crossing number of K. We show that there are sequences of non-alternating links …

We examine the conjecture, due to Champanerkar, Kofman, and Purcell that $\text {vol}(K)<
2\pi\log\det (K) $ for alternating hyperbolic links, where $\text {vol}(K)=\text {vol}(S …

We find that cusp densities of hyperbolic knots in $ S^ 3$ include a dense subset of $[0,
0.6826\dots] $ and those of links are a dense subset of $[0, 0.853\dots] $. We define a new …

Generalizing previous constructions, we present a dual pair of decompositions of the
complement of a link L into bipyramids, given any multi-crossing projection of L. When L is …