Esperantism complicates knots

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
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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I am a professor of mathematics at ETH Zürich since 2008.

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