Related papers: An alternative approach to hyperbolic structures o…
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…
We explain how to construct certain potential functions for the hyperbolic structures of a knot complement, which are closely related to the analytic functions on the deformation space of hyperbolic structures.
We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently…
In this paper we present a classical construction of the Hyperbolic structure of the complement of a link in the sense of Thurston for the particular case of the Borromean rings link. As this is nothing new, the aim of this paper is to…
To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these…
Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to non-split reduced non-2-braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show…
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating…
We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting…
Hyperbolic problems can at times be solved employing symbolic arguments. This is especially true for the construction of forward (and backward) fundamental solutions. We formulate a corresponding abstract scheme and illustrate its…
In this article we review the basics of the phasor formalism in a rigorous way, highlighting the physical motivation behind it and presenting a hyperbolic counterpart of the phasor addition formula.
This chapter from the upcoming Handbook of Knot Theory (eds. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. Along with a new elementary exposition of the…
We utilize ideal bipyramids to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections.
We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a…
Given a link in a 3-manifold such that the complement is hyperbolic, we provide two modifications to the link, called the chain move and the switch move, that preserve hyperbolicity of the complement, with only a relatively small number of…
The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.
The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every…
Let $F$ be a compact orientable surface with nonempty boundary other than a disk. Let $L$ be a link in $F \times I$ with a connected weakly prime cellular alternating projection to $F$. We provide simple conditions that determine exactly…
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based…
If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume…
We study the relationship between two concepts: cut limits and hyperbolic extensions.