Related papers: Hyperbolicity and Volume of Hyperbolic Bongles
We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows…
Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot $4_1$ and the links $5^2_1$ and $6^2_2$, have been obtained by the second named author and his…
We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the…
In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed.
We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.
In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with…
The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link…
We define a class of links in handlebodies called ``charm bracelets," which are a subset of staked links. We provide tools to construct infinitely many such hyperbolic links and bound the corresponding volumes from below in terms of volumes…
A polyhedron in a three-dimensional hyperbolic space is said to be generalized if finite, ideal and truncated vertices are admitted. In virtue of Belletti's theorem (2021) the exact upper bound for volumes of generalized hyperbolic…
In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the…
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…
The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3-manifolds, obtained by…
We show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots.
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a…
For families of knots and links given in Conway notation we compute lower maximal and upper minimal bound of hyperbolic volume by using source links and augmented links.
In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a…
In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume of a regular simplex and the $(n-1)$-dimensional volume of its facets. In addition to the…
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds.…
In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.
Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.