Table of Contents
 [[hmm that looks pretty similar… https://math.stackexchange.com/a/1792315/15108]]
 [[Visualisations for SU2, manually entangling loops]] [[think]]

[20190201]
my initial notes [[how to define manifolds? parametric? works for Rn Sn and Tn]]
 [[somehow use van kampen's theorem?]]
[20190201]
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?[20190202]
hmm. if you consider a torus, composed of two triangles, looks like all vertices are labeled in the same
[20190201]
initial python impl in file:projects/fund/main.py [[name it 'mental' (for fundamental) instead of topology? or pione??]]
 [[related]] [[topology]] [[viz]]
[20190201]
SnapPea  Wikipedia [[autopology]][20190210]
Dynamical triangulation of the 2torus  YouTube [[qg]] [[autopology]][20190510]
Keep it Simplex, Stupid!  Bartosz Milewski's Programming Cafe [[autopology]]
[[summary on trying to understand triangulated fundamental group]] [[topology]] [[autopology]]
 [[https://math.stackexchange.com/questions/1778421/fundamentalgroupofthesphereviatriangulation]] [[autopology]]
 [[http://homepage.divms.uiowa.edu/~jsimon/COURSES/M201Fall08/HandoutsAndHomework/Graph1.pdf]]
 [[klein bottle (with triangulation) https://math.stackexchange.com/questions/1778465/fundamentalgroupkleinbottletriangulation]]
 [edgepath https://en.wikipedia.org/wiki/Fundamental_group#Edgepath_group_of_a_simplicial_complex](#dgpthsnwkpdrgwkfndmntlgrpdgpthgrpfsmplclcmplx TIDDLYLINK)
 [[faces in triangulation must be distinct]]
 [[make sure you don't have loop in spanning tree]]
 [[fun fact: computing fundamental group is undecidable https://mathoverflow.net/a/304484/29889 (in terms of figuring out whether it's trivial)]]
 [https://math.stackexchange.com/questions/1666146/fundamentalgroupfromtriangulation#comment3399175_1666146](#smthstckxchngcmqstnsfndmntlgrpfrmtrngltncmmnt TIDDLYLINK)
 [[basically, I don't understand what all they mean by subcomplex. why do they color all of it???]]
 [[book by Armstrong, p. 134. Don't really understand the statement, it's all very vague]]
 [[torus – here they mention there are quite a lot of relations…]]
[20190210]
at.algebraic topology  Algorithm for computing fundamental group of simplicial complexes  MathOverflow[20190210]
at.algebraic topology  Algorithm for computing fundamental group of simplicial complexes  MathOverflow
https://en.wikipedia.org/wiki/Computational_topology
https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B8%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81
http://graphics.stanford.edu/courses/cs46809fall/
hmm wonder if that does it. they mention triangulation.
https://en.wikipedia.org/wiki/Triangulation_(topology)
https://en.wikipedia.org/wiki/Digital_manifold
hmm that looks pretty similar… https://math.stackexchange.com/a/1792315/15108
Visualisations for SU2, manually entangling loops [[think]]
Approximate them by segments, that should work
hmm. sample random loops and then try to contract them?
[20190201]
my initial notes
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?
how to define manifolds? parametric? works for Rn Sn and Tn
somehow use van kampen's theorem?
[20190201]
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?
basically you declare some of the loops as 'trivial', but manually
[20190202]
hmm. if you consider a torus, composed of two triangles, looks like all vertices are labeled in the same
however, that's not enough to classify, we should be considering edges instead?
[20190201]
initial python impl in <projects/fund/main.py>
pretty inefficient, should rewrite in rust for finer control. also, make multithreaded
name it 'mental' (for fundamental) instead of topology? or pione??
related [[topology]] [[viz]]
[20190201]
SnapPea  Wikipedia [[autopology]]
https://en.wikipedia.org/wiki/SnapPea
[20190210]
Dynamical triangulation of the 2torus  YouTube [[qg]] [[autopology]]
https://www.youtube.com/watch?v=c3NdgSIe030
[20190510]
Keep it Simplex, Stupid!  Bartosz Milewski's Programming Cafe [[autopology]]
https://bartoszmilewski.com/2018/12/11/keepitsimplexstupid/
summary on trying to understand triangulated fundamental group [[topology]] [[autopology]]
 State "START" from
[20190221]
https://math.stackexchange.com/questions/1778421/fundamentalgroupofthesphereviatriangulation [[autopology]]
FG for the sphere
http://homepage.divms.uiowa.edu/~jsimon/COURSES/M201Fall08/HandoutsAndHomework/Graph1.pdf
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??
klein bottle (with triangulation) https://math.stackexchange.com/questions/1778465/fundamentalgroupkleinbottletriangulation
edgepath https://en.wikipedia.org/wiki/Fundamental_group#Edgepath_group_of_a_simplicial_complex
faces in triangulation must be distinct
https://math.stackexchange.com/a/1772664/15108
https://math.stackexchange.com/a/954164/15108
make sure you don't have loop in spanning tree
https://math.stackexchange.com/a/1778957/15108
fun fact: computing fundamental group is undecidable https://mathoverflow.net/a/304484/29889 (in terms of figuring out whether it's trivial)
encode turing machines via topological spaces? lol
https://math.stackexchange.com/questions/1666146/fundamentalgroupfromtriangulation#comment3399175_1666146
look at remaining
basically, I don't understand what all they mean by subcomplex. why do they color all of it???
https://math.stackexchange.com/questions/2472310/findingfundamentalgroupofsimplicialcomplexes
book by Armstrong, p. 134. Don't really understand the statement, it's all very vague
torus – here they mention there are quite a lot of relations…
https://books.google.co.uk/books?id=xwzX9h_hyMUC&pg=PA202&lpg=PA202&dq=%22fundamental+group%22+triangulation+spanning+tree&source=bl&ots=m9NZ4m5lP4&sig=ACfU3U0epWUJDPHbx_RBUiq6uoTL6Zhj6Q&hl=en&sa=X&ved=2ahUKEwjhqcrXr7HgAhXwIjQIHbVmD10Q6AEwBXoECAkQAQ#v=onepage&q=%22fundamental%20group%22%20triangulation%20spanning%20tree&f=false
book: simplicial structure by ferrario, p.202
right, apparently to compute really effeciently we need van kampen theorem..
[20190210]
at.algebraic topology  Algorithm for computing fundamental group of simplicial complexes  MathOverflow
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 2simplices. I don't really see anything interesting going on algorithmically once you've selected a maximal tree.
[20190210]
at.algebraic topology  Algorithm for computing fundamental group of simplicial complexes  MathOverflow
Depends on what you mean by "computing" and "algorithm". It is undecidable (even for a twocomplex) whether the fundamental group is trivial, though computing a presentation is relatively easy.
 public document at doc.anagora.org/fgroup
 video call at meet.jit.si/fgroup
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