homology_of_vr_complexes.ipynb

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   "source": [
    "In this notebook we will demonstrate how Polymake can be used to compute a Vietoris Rips complex and its homology on a random two dimensional point cloud. To read up on these concepts we suggest to take a look at: [Computational Topology: An Introduction](https://www.researchgate.net/publication/220692408_Computational_Topology_An_Introduction) by Herbert Edelsbrunner and John Harer. \n",
    "\n",
    "We will begin by checking on our system by calling `versioninfo()`. We can see the version of our Julia Kernel and some system specs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.4.0\n",
      "Commit b8e9a9ecc6 (2020-03-21 16:36 UTC)\n",
      "Platform Info:\n",
      "  OS: Linux (x86_64-pc-linux-gnu)\n",
      "  CPU: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-8.0.1 (ORCJIT, ivybridge)\n"
     ]
    }
   ],
   "source": [
    "versioninfo()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we check the status of our package manager and tell Julia which packages to use."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1mStatus\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      " \u001b[90m [1f15a43c]\u001b[39m\u001b[37m CxxWrap v0.9.1\u001b[39m\n",
      " \u001b[90m [b4f34e82]\u001b[39m\u001b[37m Distances v0.8.2\u001b[39m\n",
      " \u001b[90m [7073ff75]\u001b[39m\u001b[37m IJulia v1.21.1\u001b[39m\n",
      " \u001b[90m [91a5bcdd]\u001b[39m\u001b[37m Plots v1.0.5\u001b[39m\n",
      " \u001b[90m [d720cf60]\u001b[39m\u001b[37m Polymake v0.4.1\u001b[39m\n",
      "polymake version 4.0\n",
      "Copyright (c) 1997-2020\n",
      "Ewgenij Gawrilow, Michael Joswig, and the polymake team\n",
      "Technische Universität Berlin, Germany\n",
      "https://polymake.org\n",
      "\n",
      "This is free software licensed under GPL; see the source for copying conditions.\n",
      "There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "using Pkg\n",
    "Pkg.status()\n",
    "\n",
    "using Polymake\n",
    "using Plots \n",
    "using Distances"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With this our setup is complete. We will begin by generating a random set of points via some of Julia's basic functionality. Then we will plot these points using the \"Plots\" package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "scrolled": true
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   "outputs": [
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    "points = rand(Float64, (16,2))\n"
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  {
   "cell_type": "code",
   "execution_count": 118,
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     "execution_count": 118,
     "metadata": {},
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   ],
   "source": [
    "scatter(points[:,1],points[:,2],xlabel=\"x-coordinate\",ylabel=\"y-coordinate\",legend=false,aspect_ratio =:equal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Polymake requires a distance matrix to compute the Vietoris rips complex. Here we use the \"Distances\" package to compute a $n \times n$ matrix of pairwise distances, where $n$ is the number of points. \n",
    "We can specify which metric we want to use and choose the Euclidean metric. \n",
    "\n",
    "Setting `dims = 1` specifies that we want the pairwise distances of the rows of our matrix `points`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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   ],
   "source": [
    "distances = Distances.pairwise(Distances.Euclidean(),points,dims = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can use Polymake's topaz library to generate the VR complex from our distance matrix. We also have to pass a float, indicating the maximum pairwise distance of vertices, such that they become a face of our abstract simplicial complex. \n",
    "Depending on our point set and the value we choose here, we get a higher or lower dimensional abstract simplicial complex. Note that if we'd choose a value of $1$ the generated abstract simplicial complex would contain all faces of the $n-$simplex."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pm::Array<pm::Set<long, pm::operations::cmp>>\n",
       "{0}\n",
       "{1 6 7}\n",
       "{1 6 13}\n",
       "{1 9 13}\n",
       "{2 8 12}\n",
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     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "vr = topaz.vietoris_rips_complex(distances, 0.33)\n",
    "vr.FACETS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Above we can see the facets, encoded by sets of labeled vertices. Below we see a \n",
    "visualization of the VR complex provided by Polymake."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
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       "\tby Ronan Oger\n",
       "\tInfo: http://www.roitsystems.com/\n",
       "\t-->\n",
       "</svg>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Polymake.display_svg(vr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we use another Polymake function to print the reduced homology of our complex. Each line corresponds to one dimension, starting with dimension $0$. The torsion coefficients are displayed in the curly brackets, which are empty if there is no torsion. The number after the curly brackets is the reduced Betti number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PropertyValue wrapping pm::Array<polymake::topaz::HomologyGroup<pm::Integer>>\n",
       "({} 1)\n",
       "({} 0)\n",
       "({} 0)\n"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vr.HOMOLOGY"
   ]
  }
 ],
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