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 knot complement”
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.
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 …
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…
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…
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…
We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orb…
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 exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.
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…
Study counts geodesic surfaces in knot complements, finding unique ones for small knots.
problem Counting totally geodesic surfaces in knot complements.
method Adapting boundary slope and intersection techniques, extending obstructions.
result Uniqueness of geodesic surfaces for specific knots, no geodesic surfaces for 47 knots.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
We establish a pair of criteria for proving that most knot complements obtained as Dehn fillings of a given two-component hyperbolic link complement lack hidden symmetries. To do this, we use certain rational functions on varieties associated to the link. We apply our criteria to show that among certain infinite famili…
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.
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.
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…
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.
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.
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.
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic …
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.
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
We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.
The paper proves rigidity of surgeries on the figure-eight knot complement.
problem Infinitesimal projective rigidity of surgeries on the figure-eight knot complement.
method Computer-assisted proof and explicit representations of the knot complement.
result Proves infinitesimal projective rigidity for surgeries far from the ideal point.
The study limits the number of 2-holed tori in knot exteriors.
problem Bounding the number of 2-holed tori in knot exteriors.
method Continuing Motegi's program, the paper applies universal bounds to hyperbolic knots.
result There are at most six non-isotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.
Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \neq 7, we show that M is the unique knot complement in its class. We include examples to illustr…
We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
The study finds infinitely many twist knot complements with totally geodesic surfaces.
problem Finding infinitely many twist knot complements with a specific number of totally geodesic surfaces.
method Using a family of twist knot complements and their dihedral covers, the authors construct examples of hyperbolic 3-manifolds with totally geodesic surfaces.
result The construction of infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces for any positive integer k.
We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large v…
We show that given n>0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order <=n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded…
Study of lambda lengths in figure eight knot complement using Eisenstein integers.
problem Determining lambda lengths in the figure eight knot complement.
method Using hyperbolic geometry and spinors, mapping lambda lengths to Eisenstein integers.
result Lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit.
Spacious knotsmath.GT
We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at le…
Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.
problem Shortest periodic geodesic on hyperbolic orbisphere with cone points.
method Computation of linking numbers to show homeomorphism.
result Lift of shortest periodic geodesic is homeomorphic to figure-eight knot complement.
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 standard ideas from Thurston's work, the article includes never-before-published ex…
Mixed-platonic 3-manifoldsmath.GT
New mixed-platonic 3-manifolds from different polyhedra types.
problem Finding new hyperbolic knot complements with hidden symmetries.
method Defined mixed-platonic 3-manifolds and studied their properties.
result No mixed-platonic hyperbolic knot complement has hidden symmetries.
Given a compact orientable 3-manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of C is as small as you like.
We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials.
Asymptotics of quantum invariants of surface diffeomorphisms II: The figure-eight knot complementmath.GT
Authors prove quantum invariant conjecture for figure-eight knot complement.
problem Connecting quantum invariants of surface diffeomorphisms to hyperbolic volumes.
method Analyzes the simplest case of a one-puncture torus and figure-eight knot complement.
result Proves conjecture linking quantum invariant to hyperbolic volume.
It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and…
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…
Study counts specific surfaces in Montesinos knots with 4 rational tangles.
problem Investigating closed essential surfaces in Montesinos knots with 4 rational tangles.
method Analyzing the number of closed, connected, essential, orientable surfaces of fixed genus in knot complements.
result Exactly 12 genus 2 surfaces and 8φ(g - 1) surfaces of genus greater than 2 are found, independent of knot crossings.
Hyperbolic Knotsmath.GT
In 1978, W. Thurston revolutionized low diemsional topology with his work on hyperbolic 3-manifolds. In this paper, we discuss what is currently known about knots in the 3-sphere with hyperbolic complements. Then focus is on geometric invariants coming out of the hyperbolic structures. This is one of a collection of ar…
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.