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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\set …
9
votes
1
answer
469
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 (th …
9
votes
2
answers
805
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 $M$, and I …
20
votes
1
answer
1k
views
Why is there a unique hyperbolic simplex of largest area?
Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$?
For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and M …
12
votes
1
answer
570
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Is there a general dilogarithm formula for the Cheeger–Chern–Simons class?
I'm looking for a generalization of the calculation of the hyperbolic volume and Chern–Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
Recall …
9
votes
1
answer
721
views
Is there a simplicial volume definition of Chern Simons invariants?
Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the fun …