[2019-02-01]
[2019-02-02]
[2019-02-10]
[2019-05-10]
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/cs468-09-fall/ hmm wonder if that does it. they mention triangulation. https://en.wikipedia.org/wiki/Triangulation_(topology) https://en.wikipedia.org/wiki/Digital_manifold 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? [2019-02-01] 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? [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? basically you declare some of the loops as ‘trivial’, but manually [2019-02-02] 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? [2019-02-01] 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]] [2019-02-01] SnapPea - Wikipedia [[autopology]] https://en.wikipedia.org/wiki/SnapPea [[qg]] [[autopology]] https://www.youtube.com/watch?v=c3NdgSIe030 [2019-05-10] Keep it Simplex, Stupid! |   Bartosz Milewski’s Programming Cafe [[autopology]] https://bartoszmilewski.com/2018/12/11/keep-it-simplex-stupid/ summary on trying to understand triangulated fundamental group [[topology]] [[autopology]] State "START" from [2019-02-21] https://math.stackexchange.com/questions/1778421/fundamental-group-of-the-sphere-via-triangulation [[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/fundamental-group-klein-bottle-triangulation edge-path https://en.wikipedia.org/wiki/Fundamental_group#Edge-path_group_of_a_simplicial_complex 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/fundamental-group-from-triangulation#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/finding-fundamental-group-of-simplicial-complexes 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.. [2019-02-10] at.algebraic topology - Algorithm for computing fundamental group of simplicial complexes - MathOverflow https://mathoverflow.net/questions/304481/algorithm-for-computing-fundamental-group-of-simplicial-complexes 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. [2019-02-10] at.algebraic topology - Algorithm for computing fundamental group of simplicial complexes - MathOverflow https://mathoverflow.net/questions/304481/algorithm-for-computing-fundamental-group-of-simplicial-complexes 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.
https://en.wikipedia.org/wiki/Triangulation_(topology)
Approximate them by segments, that should work
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?
basically you declare some of the loops as ‘trivial’, but manually
however, that’s not enough to classify, we should be considering edges instead?
pretty inefficient, should rewrite in rust for finer control. also, make multithreaded
https://en.wikipedia.org/wiki/SnapPea
https://www.youtube.com/watch?v=c3NdgSIe030
https://bartoszmilewski.com/2018/12/11/keep-it-simplex-stupid/
[2019-02-21]
FG for the sphere
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/a/1778957/15108
look at remaining
https://math.stackexchange.com/questions/2472310/finding-fundamental-group-of-simplicial-complexes
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..
https://mathoverflow.net/questions/304481/algorithm-for-computing-fundamental-group-of-simplicial-complexes
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.
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.