All Questions
Tagged with hyperbolic-geometry knot-theory
7 questions with no upvoted or accepted answers
10
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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
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331
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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
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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
4
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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
2
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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 $\pi(K) = \pi_1(S^3 \setminus K)$...
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2
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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
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213
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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\...
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