We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.
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48 results for “Hyperbolic complements”
The paper classifies cusp types in hyperbolic knot complements.
problem Classifying cusp types in hyperbolic knot complements.
method Analyzing quotients of hyperbolic knot complements.
result All cusp types arise in the quotients of link complements.
Preserves hyperbolicity in link complements with two moves.
problem Maintaining hyperbolicity in link complements.
method Chain move and switch move to preserve hyperbolicity.
result Preserves hyperbolicity in a substantial number of links.
Study shows most hyperbolic knot complements lack hidden symmetries.
problem Proving most knot complements lack hidden symmetries.
method Using rational functions on varieties associated to a link.
result Most knot complements lack hidden symmetries.
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
In this paper, we show that any non-arithmetic hyperbolic -bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic -bridge link complement cannot irregularly cover a hyperbolic -manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a ch…
Anosov flows' orbit complements are hyperbolic manifolds.
problem Characterizing hyperbolicity of Anosov flow orbit complements.
method Analyzing foliations and periodic orbits in 3D manifolds.
result Complement of any filling periodic orbit in Anosov flows is hyperbolic.
Upper and lower bounds for hyperbolic rod complements in 3-torus volumes.
problem Understanding geometric properties of hyperbolic rod complements in 3-torus.
method Provided upper and lower bounds for volumes in terms of rod parameters.
result Volume bounds for hyperbolic rod complements in 3-torus depend on rod parameters.
The Exact Computation of A Real Hyperbolic Structure on the Complement of the Borromean Ringsmath.GT
Paper computes hyperbolic structure of Borromean rings complement.
problem Computing hyperbolic structures for link complements.
method Classical construction of Thurston's sense.
result Exact computation of hyperbolic structure for Borromean rings.
Study proves hyperbolic structures for link complements in Seifert fibered spaces.
problem Proving hyperbolic structures for link complements in Seifert fibered spaces.
method Combinatorial bounds on volume of hyperbolic structures.
result Complement of a link in a Seifert fibered space admits a hyperbolic structure of finite volume.
We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with clo…
Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.
problem Finding bounds on the length of shortest closed geodesics in hyperbolic link complements.
method Established an upper bound for the length of an nth shortest closed geodesic as a logarithmic function of the volume of the manifold.
result An upper bound of the length of an nth shortest closed geodesic is established as a logarithmic function of the volume of the manifold.
3-manifolds with hyperbolic handlebody complements are studied.
problem Understanding the geometry of 3-manifolds with hyperbolic handlebody complements.
method Capping off spherical and torus boundaries, constructing handlebodies, and applying octahedral decomposition.
result Bounds on volume for some handlebody complements are derived.
Profinite rigidity proven for many hyperbolic manifolds.
problem Profinite rigidity of hyperbolic manifolds.
method Geometric topology and bubble-drilling construction.
result Profinite rigidity of many cusped hyperbolic manifolds.
We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain ba…
The paper proves limitations on hyperbolic knot complements with hidden symmetries.
problem Identifying hyperbolic knot complements with hidden symmetries.
method Obstructions to infinitely many fillings producing knot complements with hidden symmetries.
result The figure-eight knot complement is the only hyperbolic knot complement with hidden symmetries.
Study shows arithmetic properties of specific hyperbolic Dehn fillings.
problem Arithmeticity of one-cusped Dehn fillings of specific link complements.
method Investigation of cusp fields, trace fields, and invariant trace fields.
result No one-cusped hyperbolic Dehn filling of the Berge manifold is arithmetic.
We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-ma…
We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric.
We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
Study volume conjecture for links with multiple hyperbolic pieces.
problem Volume conjecture for links with more than one hyperbolic piece.
method Constructing infinite families of prime links, analyzing their complements, and using colored Jones polynomials and simplicial volume.
result Exponential growth rates of colored Jones polynomials capture the simplicial volume of link complements.
Hyperbolic knots decompose into prism orbifolds.
problem Understanding hyperbolic knot complements and their geometric properties.
method Analyzing knot complements as quotients of by discrete groups of reflections in polyhedra with triangular prism combinatorial type.
result Knot complements decompose into hidden symmetries and contain closed, embedded, totally geodesic surfaces.
We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic…
We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most hyperbolic knot complements in a cyclic commensurability class. Moreover …
Researchers found only one hyperbolic structure for Borromean rings.
problem Characterizing hyperbolic structures in knot complements.
method Analyzing the fundamental group of the Borromean rings' complement and its representations in PSL(2,C).
result Borromean rings admit exactly one hyperbolic structure.
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 construct examples of codimension two hyperbolic link complements in closed smooth 4-manifolds with homeomorphism type . All our examples are based on a construction of J. Ratcliffe and S. Tschantz, who constructed 1171 non-compact finite volume hyperbolic 4-manifolds of minimal volume. We the…
Many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. Generalizing to dimension 4, we construct a dozen examples of noncompact hyperbolic 4-manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of some o…
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp sha…
It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to…
Study on colored Jones polynomial and link complements.
problem Understanding the structure of link complements with arbitrary colors.
method Investigated the potential function of the colored Jones polynomial and established a relationship with hyperbolicity.
result Evidence supports the Chen-Yang conjecture on link complements.
Geometrically describes hyperbolic structures on link complements using quantum groups.
problem Describing hyperbolic structures on link complements algebraically.
method Uses octahedral decomposition and Kashaev-Reshetikhin's braiding on quantum group .
result Shows how to interpret geometrically the algebraic gluing equations for hyperbolic structures.
The paper identifies knots in specific lens spaces based on their complements.
problem Determining knots in homology lens spaces based on their complements.
method Analyzing knots in homology lens spaces with specific conditions.
result Knots are determined by their complements under certain conditions.
This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifo…
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 Dehn filling on the figure eight knot complement, that are uniquely determined by …
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. The 3-manifold is the complement of a link with eight components, and its volume…
Study calculates quantum hyperbolic invariants for figure-eight knot complement, finding it either 0 or half the volume.
problem Computing quantum hyperbolic invariants for knot complements.
method Computed the real part of the semi-classical limit of quantum hyperbolic invariants of the figure-eight knot complement.
result The real part is rigid and either 0 or half the hyperbolic volume of the knot complement.
An alternative method is described for determining the hyperbolic structure on a link complement, and some of its elementary consequences are examined. The method is particularly suited to alternating links.
This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified
The study finds a bound on the shortest geodesics in hyperbolic 3-manifolds.
problem Finding bounds on the shortest geodesics in hyperbolic 3-manifolds.
method Establishing an upper bound for the length of the shortest closed geodesic in terms of the volume of the manifold.
result An upper bound for the length of the shortest closed geodesic in terms of the volume of the manifold.
Profinite rigidity of certain 3-manifolds detected through Dehn fillings.
problem Detecting profinite rigidity in 3-manifolds.
method Observation of profinite isomorphisms between cusped hyperbolic 3-manifolds and their Dehn fillings.
result Some cusped hyperbolic 3-manifolds are profinitely rigid.
In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the -sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid con…
Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.
We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.
Highly twisted diagramsmath.GT
New proof shows certain knots are hyperbolic.
problem Characterizing knots with highly twisted diagrams.
method New approach involving Euler characteristic argument.
result Complements of certain knots are unannular and atoroidal.
The paper finds lower bounds on hyperbolic 3-manifold volumes.
problem Finding lower bounds on hyperbolic 3-manifold volumes.
method Decomposing a 3-manifold into hyperbolic pieces and summing their volumes.
result The volume of a 3-manifold is bounded below by the sum of the volumes of its hyperbolic pieces.
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.