All Questions
Tagged with hyperbolic-geometry knot-theory
37 questions
16
votes
5
answers
4k
views
can you fool SnapPea?
A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with.
What I'm looking for is a non-hyperbolizable knot ...
- 44.4k
14
votes
2
answers
789
views
Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?
Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
What if in Question 1 we restrict the cone angles to be $...
- 285
10
votes
0
answers
127
views
Compatibility of spherical and hyperbolic geometry for fibred knots
Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
- 44.4k
9
votes
3
answers
802
views
What constant ensures hyperbolicity of Dehn surgery?
I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...
- 1,000
9
votes
2
answers
812
views
How many knots are there with hyperbolic volume less than a given constant
Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...
- 18.7k
9
votes
1
answer
480
views
What is the complex structure on the boundary torus of a hyperbolic knot complement?
Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary (...
- 18.7k
9
votes
1
answer
1k
views
On the geometrization of double branched covers
I recently got into Lickorish's paper Prime knots and tangles and a question, which I didn't have the first time I read it, naturally emerged.
The Thurston-Perelman Geometrization Theorem asserts ...
- 606
9
votes
0
answers
331
views
Connections between spectral geometry and critical point/Morse theory
I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
- 217
8
votes
2
answers
642
views
Hyperbolic Brunnian links and rectangular cusp shapes
My question is as follows.
Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?
Here is what I mean:
The Borromean rings form a famous link $B$ (a smooth closed ...
- 1,131
8
votes
1
answer
660
views
On trivial mapping class group of 3-manifolds
What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
- 3,751
8
votes
1
answer
413
views
Does the limit in the Volume conjecture converge?
The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...
- 2,869
8
votes
0
answers
432
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
- 111
7
votes
2
answers
507
views
What is known about exceptional slopes of hyperbolic knots?
For $K$ a hyperbolic knot in $S^3$, a rational number $p/q$ is an exceptional slope if the manifold $M_{p/q}$ obtained from $(p,q)$-Dehn surgery on $K$ does not admit a hyperbolic structure.
Thurston ...
- 2,533
7
votes
1
answer
475
views
Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?
I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.
Following this paper by Christian ...
- 318
7
votes
1
answer
215
views
Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?
For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
- 2,533
6
votes
3
answers
555
views
Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
- 3,751
5
votes
2
answers
309
views
Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$
Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to
\Sigma$ of this projection is then ...
- 1,231
5
votes
1
answer
441
views
Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement
I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem ...
- 485
5
votes
1
answer
292
views
Conformal boundary and cusp of figure-8 complement
As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...
- 309
5
votes
1
answer
297
views
Hyperbolicity of twist knots
In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the ...
user160180
4
votes
1
answer
838
views
Knot complements with respect to Thurston's 8 geometries
We can put knot complements in three buckets: hyperbolic knots, satellite knots, and torus knots.
Can this classification be made with looking at the (complete) metric on the complement, if it has ...
- 1,465
4
votes
2
answers
306
views
Quadratic cusp shape
Which hyperbolic $3$-manifolds are known to have quadratic cusp shape?
Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...
- 10.4k
4
votes
1
answer
277
views
How many quadratic fields occur as trace fields of hyperbolic knot complements?
I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs $(F,...
- 2,436
4
votes
1
answer
236
views
Computation of cusp shape from vertex invariants
Following Takahashi ("On the concrete construction of hyperbolic structure of 3-manifolds"), I was able to construct the Euclidean cusp cross-section for the 5_2 knot complement (please see ...
- 43
4
votes
0
answers
88
views
What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?
Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
- 2,533
3
votes
2
answers
344
views
Hyperbolic volume of hyperbolic knots
Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...
- 223
3
votes
2
answers
950
views
hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
...
- 1,629
3
votes
1
answer
496
views
Classifying links with essential annuli in the complement as torus links
I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and ...
- 33
3
votes
1
answer
375
views
Chern-Simons invariants of 2-bridge knots
2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below:
Looking at some ...
- 10.4k
2
votes
1
answer
645
views
Is there a criterion for a link complement to have a hyperbolic structure with finite volume
For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with ...
- 957
2
votes
2
answers
302
views
Complement of figure 8 knot - zero vertex [closed]
Thurston in "3-dimensional Geometry and Topology" explicitely creates the hyperbolic complement of the figure 8 knot by glueing two ideal tetrahedra.
What I do not understand is that he pushes the ...
- 1,465
2
votes
0
answers
162
views
Can distinct meridians commute in a knot group?
Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...
- 2,533
2
votes
0
answers
60
views
Relation between symmetries of hyperbolic knot and the symmetries of a generic triangulation
For canonical ideal triangulation of a hyperbolic knot, the symmetries of the knot are the same as the symmetries of that triangulation. This is how SnapPy computes the knot symmetry group.
Is there ...
- 63
1
vote
1
answer
91
views
When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
- 605
1
vote
0
answers
213
views
Measuring the complexity of a knot by minimum number of simplices to tile the complement
This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement
Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb S^3\...
- 18.7k
0
votes
1
answer
127
views
What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?
I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (...
- 318
-2
votes
1
answer
141
views
Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \Gamma$, where $\Gamma$ has no parabolic generators?
Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret ...
- 111