Esperantism complicates knots – E. Kowalski's blog

One of the results in the new preprint of Gromov and Guth that I mentioned in my previous post is a rather striking application of expanders to a problem of knot theory: a new proof of the fact (only recently proved by Pardon) that there exist knots $K$ (in $\mathbf{R}^3$ ) with arbitrarily large distortion $d(K)$ , where the latter is defined by
$d(K)=\inf_{k}\sup_{x,y \in k} \frac{d_k(x,y)}{|x-y|},$
where $k$ runs over knots isotopic to $K$ , and the distance in $k$ in the numerator being the distance between points on the curve that “is” $k$ . In other words, the distortion is large when, whichever way we put the knot in space, there are points on it which are “physically close” in the ambient space, but far away if one is forced to connect them along the knot itself.

(Here and henceforth, any knot-theoretic blunders are obviously my own unenlightened ones…)

Gromov had asked whether the distortion is unbounded in the early 80’s, and Pardon gave the first examples (there is a blog post here that explains his work).

The examples of Gromov and Guth (Section 4 of their paper, which they describe as “the most interesting”…) arise from the following context: let $M$ be a fixed hyperbolic arithmetic 3-manifold; let $M_i\rightarrow M$ be any sequence of arithmetic coverings of $M$ with degree $d_i$ tending to infinity as $i$ does; then, by results of Hilden and Montesinos from the 1970’s, there are knots $K_i$ associated to $M_i$ by the property that $M_i$ can be written as a $3$ -cover of the sphere $\mathbf{S}^3$ ramified over $K_i$ . Then Gromov and Guth prove that
$d(K_i)\gg h_i V_i,$
where $h_i$ is the Cheeger constant for $M_i$ , and
$V_i=d_i\mathrm{Vol}(M)$
is the hyperbolic volume of $M_i$ .

Now, by the Cheeger-Buser principle, we have
$h_i\gg \lambda_1(M_i),$
where $\lambda_1(M_i)$ is the first positive eigenvalue of the hyperbolic Laplace operator on $M_i$ . By Property (τ) for $M$ — which uses the arithmetic property of the coverings –, this first eigenvalue is bounded away from zero, and hence the Gromov-Guth inequality gives
$\lim_{i\rightarrow +\infty} d(K_i)=+\infty.$

The link with expanders is the Brooks-Burger comparison principle: the $\lambda_1$ of the hyperbolic Laplace operator is (up to constants depending only on $M$ and a choice of generators of its fundamental group) of the same order of magnitude as the first positive eigenvalue for the Cayley-Schreier graphs $\Gamma_i$ associated to the covering $M_i\rightarrow M$ .

This type of setting may remind (very) faithful readers of the discussion of some of my own results with J. Ellenberg and C. Hall, where expanders turned out to be slightly overkill, a weaker degree of expansion of the graphs — which we called “esperantism” — being sufficient for our purposes.

I think — but I may have missed some problem! — that for the purpose of constructing knots with large distortion, the same basic idea also applies. Namely, if we take for $M_i$ some congruence coverings with degree $d_i\asymp i^k$ for some fixed $k>0$ , an inequality of Brooks shows that
$h_i\gg h(\Gamma_i),$
where $h(\Gamma_i)$ is the combinatorial Cheeger constant. The analogue of Cheeger-Buser for graphs leads to
$h(\Gamma_i)\gg \lambda_1(\Gamma_i),$
and if we know the esperanto property
$\lambda_1(\Gamma_i)\gg 1/(\log d_i)^A$
for some fixed $A\geq 0$ , we see that the logarithmic decay of the Cheeger constant is more than compensated by the growth of the volume in
$d(K_i)\gg h_i V_i.$

(Note: the proof of the main inequality (*) in the paper of Gromov and Guth is quite delicate and is much deeper and more sophisticated mathematics than the rather formal considerations above! I hope to be able to understand more of it in the coming weeks and months — indeed, the amount and variety of ideas and tools in this paper is quite remarkable…)

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Kowalski