autopology

# Vizualizing fundamental group

http://graphics.stanford.edu/courses/cs468-09-fall/
hmm wonder if that does it. they mention triangulation.

## ¶STRT[A] Visualisations for SU2, manually entangling loops autopologythink

CREATED: [2019-01-30]

Approximate them by segments, that should work

## ¶[2019-02-01] my initial notes autopology

Triangulate the manifold
then do some sort of random walk and stop at the initial point with some probability
then, do some sort of simulated annealing to transform the loops according to the certain rules.
basically, we can contract certain subpaths (or expand?) e.g. a -> b -> a can be contracted to a. unless the points are glued?
try to guess groups from equivalence classes? then try combining them and guessing against known groups?

to contract, define some sort of 'tension' function? not sure if makes sense

this is a conservative method in the sense that it can answer what your fundamental group is NOT
in a sense, fundamental group is a conservative concept too, it can answer what your topological space is NOT

triangulation, or squares?

if the point or edge is glued, treat it as special?

### ¶TODO[2019-02-01] mm. ok this ended up a bit different from what I imagined initially. still though, maybe contracting loops manually is ok, people are better at that? autopology

basically you declare some of the loops as 'trivial', but manually

### ¶TODO[2019-02-02] hmm. if you consider a torus, composed of two triangles, looks like all vertices are labeled in the same autopology

however, that's not enough to classify, we should be considering edges instead?

## ¶DONE[2019-02-01] initial python impl in projects/fund/main.pyautopology

pretty inefficient, should rewrite in rust for finer control. also, make multithreaded

## ¶STRT[C] summary on trying to understand triangulated fundamental group autopologytopology

CREATED: [2019-02-10]

### ¶https://math.stackexchange.com/questions/1778421/fundamental-group-of-the-sphere-via-triangulationautopologytopology

FG for the sphere

### ¶TODO[B]http://homepage.divms.uiowa.edu/~jsimon/COURSES/M201Fall08/HandoutsAndHomework/Graph1.pdfautopologytopology

most useful so far.. the idea is you construct spanning tree, choose a base point and assign all loops from base point to 1 (for each edge not in the maximal tree). does that work for higher dimensions??

### ¶https://math.stackexchange.com/questions/1666146/fundamental-group-from-triangulation#comment3399175_1666146autopologytopology

look at remaining

## ¶TODO[D][2019-02-10] at.algebraic topology - Algorithm for computing fundamental group of simplicial complexes - MathOverflow autopology

```Kruskal's algorithm will give you a maximal tree, and after that the presentation just involves listing the remaining edges as generators, and listing the relations that come from the 2-simplices. I don't really see anything interesting going on algorithmically once you've selected a maximal tree.
```

## ¶[D][2019-02-10] at.algebraic topology - Algorithm for computing fundamental group of simplicial complexes - MathOverflow autopology

```Depends on what you mean by "computing" and "algorithm". It is undecidable (even for a two-complex) whether the fundamental group is trivial, though computing a presentation is relatively easy.
```