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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 …
John Pardon's user avatar
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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 …
John Pardon's user avatar
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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 …
John Pardon's user avatar
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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 …
John Pardon's user avatar
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12 votes
1 answer
570 views

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 …
John Pardon's user avatar
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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 …
John Pardon's user avatar
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