This is a super nice subject. I happen to have done my Phd on that very subject 🙂

It’s nice to see it exposed that well.

Here are a few papers that I wrote some ten years ago that might interest you.

In that paper I gave a fast algorithm to generate all the examples :

An Optimal Algorithm to Generate Pointed Trivalent Diagrams and Pointed Triangular Maps

S. Vidal (2010)

https://arxiv.org/abs/0706.0969

Published in Theoretical Computer Science Volume 411, Issues 31–33, 28 June 2010, Pages 2945–2967

The original article: contains tables of examples and the counting in the labelled case (you previous post) and unlabelled case (up to isomorphism)

Sur la classification et le dénombrement des sous-groupes du groupe modulaire et leurs classes de conjugaison

(On the classification and counting of the subgroups of the modular group and their conjugacy classes)

S. Vidal (2007)

https://arxiv.org/pdf/math/0702223.pdf

(Unpublished)

Same thing abridged and in english : contains also a facinating connection with the asymptotics of the Airy function.

In here I did a drawing of the platonic solids in the following paper in 3d (p.5) for n=3,4,5 as you suggested.

Also I mention Klein’s quartic that arise for n=7. The case n=6 is a torus paved with hexagones if I remember correctly.

Trivalent diagrams, modular group and triangular maps

S. Vidal (2008)

http://cafemath.fr/articles/stacs_vidal_08.pdf

(Unpublished)

Related : develops the connection with the the asymptotics of the Airy function in more details

Counting Rooted and Unrooted Triangular Maps

S. Vidal, M. Petitot.

http://cafemath.fr/articles/VidalPetitot09-1.pdf

Published in Journal of Nonlinear Systems and Applications. Volume 1, Number 1-2, Pages 51-57, 2010.

What are some examples which demonstrate this philosophy? I’m guessing one of them is HH(F_p) vs THH(F_p); are there others?

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