orbit_space_simplicial_homology.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 4\n",
    "N=binomial(n,2)\n",
    "#N=4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "#list of unordered pairs of elements in {1,...,n}\n",
    "def pairs(n): return flatten([[[i+1,j+1] for j in range(i+1,n)] for i in range(n-1)],max_level=1)\n",
    "\n",
    "#sigma acts on an unordered pair\n",
    "def act(sigma, p): return sorted([sigma(p[0]),sigma(p[1])])\n",
    "\n",
    "#sigma acts as matrix in basis on a vector v\n",
    "def act_vect(g,v,basis=identity_matrix(N)): return (basis.inverse()*g.matrix()*basis)*vector(v)\n",
    "\n",
    "#permutation in Sigma_N which induced from perm in Sigma_n\n",
    "def embed(sigma,n): return perm_from_sort([act(sigma,p) for p in pairs(n)])\n",
    "\n",
    "#find permutation which orders a list L\t\n",
    "def perm_from_sort(L): return Permutation([pair[0] for pair in sorted(enumerate(L, 1), key=lambda x: x[1])]).to_cycles()\n",
    "\n",
    "#image of generators of Sigma_n under embedding into Sigma_N\n",
    "def embed_gens(n): return [embed(gen,n) for gen in SymmetricGroup(n).gens()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def trans(n): return [SymmetricGroup(n)([(i,i+1)]) for i in range(1,n)]\n",
    "def embedded_trans(n): return [SymmetricGroup(binomial(n,2))(embed(gen,n)) for gen in trans(n)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1,2), (2,3), (3,4)]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trans(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(2,4)(3,5), (1,2)(5,6), (2,3)(4,5)]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "embedded_trans(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "#return n when N=(n choose 2)\n",
    "def invert_binomial(N):\n",
    "    for n in range(floor(sqrt(2*N)),floor(sqrt(2*N)+2)):\n",
    "        if binomial(n,2) == N:\n",
    "            return n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "#naive minimum finding acting over entire group\n",
    "def find_min(x,y):\n",
    "    assert len(x) == len(y)\n",
    "    print(angle(x,y))\n",
    "    n = invert_binomial(len(x))\n",
    "    G = PermutationGroup(embed_gens(n))\n",
    "    for sigma in G:\n",
    "        if vector(x)*(sigma.matrix()*vector(y)) >= vector(x)*vector(y):\n",
    "            y = sigma.matrix()*vector(y)\n",
    "    print(angle(x,y))\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def angle(x,y): return float(vector(x)*vector(y)/(vector(x).norm()*vector(y).norm()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "#fundamental domain given arbitrary distinct vector l and basis B\n",
    "def fund_domain(center=[i for i in range(N)],basis=identity_matrix(N),br=QQ,group=MatrixGroup(identity_matrix(N))):\n",
    "\n",
    "    #augmented matrix for half-plane ineqs\n",
    "    A = [ [center[j] - act_vect(group[i],vector(center),basis)[j] for j in range(N)] for i in range(group.order())]\n",
    "    b_1 = group.order()*[0]\n",
    "    bA = matrix(br,b_1).transpose().augment(matrix(br,A))\n",
    "\n",
    "    #augmented matrix for positivity ineqs\n",
    "    pos = basis.inverse().transpose()\n",
    "    b_2 = [0 for i in range(N)]\n",
    "    bPos = matrix(br,b_2).transpose().augment(pos)\n",
    "\n",
    "    #augmented matrix for fund. domain\n",
    "    aug_l = bA.stack(bPos)\n",
    "    \n",
    "    #find convex polyhedral region, i.e. intersection of all ineqs\n",
    "    poly = Polyhedron(ieqs=aug_l,base_ring=br)\n",
    "\n",
    "    return([aug_l,poly])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "#slice the fundamental domain with the plane x_1 + ... + x_N = 1\n",
    "def cross_section(region,slice_dir=[1 for i in range(N)],chi=1,br=None):\n",
    "    if br is None: br=region.base_ring()\n",
    "    return Polyhedron(ieqs=region.inequalities(),eqns=[[-1]+slice_dir],base_ring=br)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "#find fixed points under G in polyhedron F to include as new vertices\n",
    "#find intersection of fixed point subspace with F as polyhedron, and return those vertices\n",
    "def fixed_verts(F,br=QQ):\n",
    "    new_verts=set()\n",
    "    for g in G:\n",
    "        #get equations for fixed point subspace\n",
    "        A=g.matrix() #matrix associated to group element g\n",
    "        B=A-identity_matrix(N) #eqns defining fixed pt subspace are (A-Id)x == 0\n",
    "        b = N*[0] #0 vector\n",
    "        bB = matrix(br,b).transpose().augment(matrix(br,B))#form augmented matrix for equations defining subspace\n",
    "        #get equations for intersection\n",
    "        eqns=matrix(br,[list(eq) for eq in F.equations()]) #list of equations defining F\n",
    "        eqns=eqns.stack(bB) #augmented matrix including fixed point subspace equations\n",
    "        #get list of inequalities for fundamental domain as convex polyhedron\n",
    "        ieqs=matrix(br,[list(ieq) for ieq in F.inequalities()])\n",
    "        #form intersection of fund_domain with fixed point subspace\n",
    "        intersection_F_fixed_pt_subspace=Polyhedron(ieqs=ieqs,eqns=eqns)\n",
    "        verts=intersection_F_fixed_pt_subspace.vertices()\n",
    "        for vert in verts:\n",
    "            new_verts.add(tuple(vert))\n",
    "    return sorted(list(new_verts))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "#action of group element g on a face\n",
    "#if g.face is not in fund_domain defined by vertex list, return None\n",
    "def act_face(g,face,vertices):\n",
    "    try:\n",
    "        return [vertices.index(tuple(act_vect(g,vertices[i]))) for i in face]\n",
    "    except ValueError:\n",
    "        return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "#determine if face1 is glued to face2 by the action of G\n",
    "#faces are given by a list vertex indices\n",
    "def faces_glued(face1,face2,G,vertices):\n",
    "    for g in G:\n",
    "        g_face1 = act_face(g,face1,vertices)\n",
    "        if g_face1 is not None and set(g_face1) == set(face2):\n",
    "            return True\n",
    "    return False"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "#for two glued faces face1, face2, determine if the gluing map preserves orientation\n",
    "#if a gluing map g1 preserves orientation, then all do since g1*g2^(-1) is a self map and must be trivial\n",
    "def gluing_preserves_orientation(face1,face2,vertices):\n",
    "    S=SymmetricGroup(range(len(vertices))) #symmetric group on vertex indices\n",
    "    for g in G:\n",
    "        g_face1 = act_face(g,face1,vertices)\n",
    "        if g_face1 is not None and set(g_face1) == set(face2):\n",
    "            sigma_g=perm_from_sort(g_face1)\n",
    "            g_cycle=PermutationGroupElement(sigma_g,parent=S) #get permutation which orders g.v_{i_0}...g.v_{i_k}\n",
    "            return g_cycle.sign()\n",
    "    return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "#determine if there is group element which glues face non-trivially to itself\n",
    "def trivial_self_gluing(face,G,vertices):\n",
    "    trivial = True\n",
    "    for g in G:\n",
    "        #act on vertices defining face\n",
    "        g_face = act_face(g,face,vertices)\n",
    "        if g_face is not None and set(g_face) == set(face):\n",
    "            #check if the gluing is non-trivial\n",
    "            if [g_face.index(i) for i in face] != [i for i in range(len(face))]:\n",
    "                trivial = False\n",
    "    return trivial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "#toss out any facets with non-trivial self-gluings such as [0,3,4] or [1,6]\n",
    "#build list of faces for each dim\n",
    "import itertools\n",
    "def faces(k): return [face for face in itertools.combinations(range(len(vertices)), k+1) if trivial_self_gluing(face,G,vertices)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "#find classes of vertices which are glued\n",
    "def glued_faces(k):\n",
    "    glued_verts = []\n",
    "    for i in faces(k):\n",
    "        found = False\n",
    "        for equiv_class in glued_verts:\n",
    "            if len(equiv_class) >= 1:\n",
    "                if faces_glued(i,equiv_class[0],G,vertices):\n",
    "                    equiv_class.append(i)\n",
    "                    found=True\n",
    "        if not found:\n",
    "            glued_verts.append([i])\n",
    "    return glued_verts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "#compute boundary of each face. keep track of orientation\n",
    "def boundary(face,glued_faces,vertices):\n",
    "    dim = len(face)-1 #face/simplex dimension is number of vertices in face-1\n",
    "    boundary = [0]*len(glued_faces[dim-1]) #initialize vector to count occurence boundary faces\n",
    "    for i in range(len(face)):\n",
    "        face_remove_i = face[:i] + face[i+1:] #remove vertex at index i\n",
    "        sign = (-1)^i\n",
    "        #find representative to which face\\{i} is glued to\n",
    "        for glued_face in glued_faces[dim-1]:\n",
    "            face_rep = glued_face[0] #use first face in glued_face list as representative\n",
    "            face_index = None\n",
    "            if face_remove_i in glued_face: #check if face_remove_i is in gluing class\n",
    "                face_index = glued_faces[dim-1].index(glued_face) #get index of face in glued (k-1)-faces\n",
    "                orient_preserve = gluing_preserves_orientation(face_rep,face_remove_i,vertices) #determine if gluing map preserves orientation\n",
    "            if face_index is not None:\n",
    "                boundary[face_index] += orient_preserve*sign #add up using index\n",
    "    return boundary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "#compute the boundary map matrix over \\ZZ\n",
    "def boundary_map(k,glued_face_list,vertices): return matrix(ZZ,[boundary(face[0],glued_face_list,vertices) for face in glued_face_list[k]]).transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "#define a chain complex, optionally truncated at max_degree\n",
    "def chain_complex(max_degree=1): return ChainComplex({k:boundary_map(k,glued_face_list,vertices) for k in range(1,max_degree+1)},degree=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "#define the finite group\n",
    "G = PermutationGroup(embed_gens(n)) #symmetric group \\sigma_n as a subgroup of \\sigma_N, with 2 generators\n",
    "#G = MatrixGroup(matrix(QQ,[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]))\n",
    "#G=SymmetricGroup(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A 6-dimensional polyhedron in QQ^6 defined as the convex hull of 1 vertex and 16 rays (use the .plot() method to plot)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#compute fundamental domain to get list of vertices\n",
    "F=fund_domain(group=G,center=[1,2,3,5,6,8]); F[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0, 0, 0, 1/3, 1/3, 1/3),\n",
       " (0, 0, 1/3, 0, 1/3, 1/3),\n",
       " (0, 0, 9/22, 25/66, 7/66, 7/66),\n",
       " (25/66, 0, 7/66, 0, 7/66, 9/22),\n",
       " (2/9, 0, 2/9, 2/9, 1/9, 2/9),\n",
       " (9/22, 0, 0, 7/66, 7/66, 25/66),\n",
       " (1/4, 0, 1/4, 1/4, 0, 1/4),\n",
       " (1/6, 1/6, 1/6, 1/6, 1/6, 1/6),\n",
       " (1/2, 0, 0, 0, 0, 1/2),\n",
       " (1/4, 1/4, 0, 0, 1/4, 1/4),\n",
       " (2/5, 0, 0, 0, 1/5, 2/5),\n",
       " (0, 0, 0, 0, 1/2, 1/2),\n",
       " (0, 2/5, 0, 0, 2/5, 1/5),\n",
       " (0, 0, 0, 0, 0, 1),\n",
       " (0, 2/9, 2/9, 2/9, 2/9, 1/9),\n",
       " (0, 0, 2/5, 2/5, 0, 1/5)]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#compute a cross section of the fundamental domain (sum of components is constant)\n",
    "F2=cross_section(F[1])\n",
    "[tuple(v) for v in F2.vertices()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0, 0, 0, 0, 0, 1),\n",
       " (0, 0, 0, 0, 1/2, 1/2),\n",
       " (0, 0, 0, 1/3, 1/3, 1/3),\n",
       " (0, 0, 1/3, 0, 1/3, 1/3),\n",
       " (0, 0, 2/5, 2/5, 0, 1/5),\n",
       " (0, 0, 9/22, 25/66, 7/66, 7/66),\n",
       " (0, 2/9, 2/9, 2/9, 2/9, 1/9),\n",
       " (0, 2/5, 0, 0, 2/5, 1/5),\n",
       " (1/6, 1/6, 1/6, 1/6, 1/6, 1/6),\n",
       " (2/9, 0, 2/9, 2/9, 1/9, 2/9),\n",
       " (1/4, 0, 1/4, 1/4, 0, 1/4),\n",
       " (1/4, 1/4, 0, 0, 1/4, 1/4),\n",
       " (25/66, 0, 7/66, 0, 7/66, 9/22),\n",
       " (2/5, 0, 0, 0, 1/5, 2/5),\n",
       " (9/22, 0, 0, 7/66, 7/66, 25/66),\n",
       " (1/2, 0, 0, 0, 0, 1/2)]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vertices=fixed_verts(F2,QQ); vertices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[(0,)],\n",
       " [(1,)],\n",
       " [(2,)],\n",
       " [(3,)],\n",
       " [(4,), (7,), (13,)],\n",
       " [(5,), (12,), (14,)],\n",
       " [(6,), (9,)],\n",
       " [(8,)],\n",
       " [(10,), (11,)],\n",
       " [(15,)]]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#compute glued 0-faces\n",
    "zero_faces=glued_faces(0); zero_faces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[(0, 1)],\n",
       " [(0, 2)],\n",
       " [(0, 3)],\n",
       " [(0, 4), (0, 7)],\n",
       " [(0, 5)],\n",
       " [(0, 6)],\n",
       " [(0, 8)],\n",
       " [(0, 9)],\n",
       " [(0, 10), (0, 11)],\n",
       " [(0, 12)],\n",
       " [(0, 13)],\n",
       " [(0, 14)],\n",
       " [(0, 15)],\n",
       " [(1, 2)],\n",
       " [(1, 3)],\n",
       " [(1, 4)],\n",
       " [(1, 5)],\n",
       " [(1, 6), (1, 9)],\n",
       " [(1, 7), (1, 13)],\n",
       " [(1, 8)],\n",
       " [(1, 10)],\n",
       " [(1, 11)],\n",
       " [(1, 12)],\n",
       " [(1, 14)],\n",
       " [(1, 15)],\n",
       " [(2, 3)],\n",
       " [(2, 4), (2, 7), (2, 13)],\n",
       " [(2, 5), (2, 14)],\n",
       " [(2, 6), (2, 9)],\n",
       " [(2, 8)],\n",
       " [(2, 10), (2, 11)],\n",
       " [(2, 12)],\n",
       " [(2, 15)],\n",
       " [(3, 4), (3, 7), (3, 13)],\n",
       " [(3, 5), (3, 12)],\n",
       " [(3, 6), (3, 9)],\n",
       " [(3, 8)],\n",
       " [(3, 10), (3, 11)],\n",
       " [(3, 14)],\n",
       " [(3, 15)],\n",
       " [(4, 5), (12, 13), (13, 14)],\n",
       " [(4, 6), (6, 7), (9, 13)],\n",
       " [(4, 8), (7, 8), (8, 13)],\n",
       " [(4, 9)],\n",
       " [(4, 10), (7, 11), (11, 13)],\n",
       " [(4, 11), (7, 10)],\n",
       " [(4, 12), (7, 12)],\n",
       " [(4, 13)],\n",
       " [(4, 14), (7, 14)],\n",
       " [(4, 15), (7, 15)],\n",
       " [(5, 6), (5, 9), (9, 12), (9, 14)],\n",
       " [(5, 7), (5, 13)],\n",
       " [(5, 8), (8, 12), (8, 14)],\n",
       " [(5, 10), (10, 12), (10, 14), (11, 12), (11, 14)],\n",
       " [(5, 11)],\n",
       " [(5, 15)],\n",
       " [(6, 8), (8, 9)],\n",
       " [(6, 10), (6, 11), (9, 11)],\n",
       " [(6, 12)],\n",
       " [(6, 13), (7, 9)],\n",
       " [(6, 14)],\n",
       " [(6, 15)],\n",
       " [(8, 10), (8, 11)],\n",
       " [(8, 15)],\n",
       " [(9, 10)],\n",
       " [(9, 15)],\n",
       " [(10, 13)],\n",
       " [(10, 15), (11, 15)],\n",
       " [(12, 15), (14, 15)],\n",
       " [(13, 15)]]"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#compute glued 1-faces\n",
    "one_faces=glued_faces(1); one_faces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[(0, 1, 2)],\n",
       " [(0, 1, 3)],\n",
       " [(0, 1, 4)],\n",
       " [(0, 1, 5)],\n",
       " [(0, 1, 6)],\n",
       " [(0, 1, 7)],\n",
       " [(0, 1, 8)],\n",
       " [(0, 1, 9)],\n",
       " [(0, 1, 10)],\n",
       " [(0, 1, 11)],\n",
       " [(0, 1, 12)],\n",
       " [(0, 1, 13)],\n",
       " [(0, 1, 14)],\n",
       " [(0, 1, 15)],\n",
       " [(0, 2, 3)],\n",
       " [(0, 2, 4), (0, 2, 7)],\n",
       " [(0, 2, 5)],\n",
       " [(0, 2, 6)],\n",
       " [(0, 2, 8)],\n",
       " [(0, 2, 9)],\n",
       " [(0, 2, 10), (0, 2, 11)],\n",
       " [(0, 2, 12)],\n",
       " [(0, 2, 13)],\n",
       " [(0, 2, 14)],\n",
       " [(0, 2, 15)],\n",
       " [(0, 3, 4), (0, 3, 7)],\n",
       " [(0, 3, 5)],\n",
       " [(0, 3, 6)],\n",
       " [(0, 3, 8)],\n",
       " [(0, 3, 9)],\n",
       " [(0, 3, 10), (0, 3, 11)],\n",
       " [(0, 3, 12)],\n",
       " [(0, 3, 13)],\n",
       " [(0, 3, 14)],\n",
       " [(0, 3, 15)],\n",
       " [(0, 4, 5)],\n",
       " [(0, 4, 6), (0, 6, 7)],\n",
       " [(0, 4, 8), (0, 7, 8)],\n",
       " [(0, 4, 9)],\n",
       " [(0, 4, 10), (0, 7, 11)],\n",
       " [(0, 4, 11), (0, 7, 10)],\n",
       " [(0, 4, 12), (0, 7, 12)],\n",
       " [(0, 4, 13)],\n",
       " [(0, 4, 14), (0, 7, 14)],\n",
       " [(0, 4, 15), (0, 7, 15)],\n",
       " [(0, 5, 6)],\n",
       " [(0, 5, 7)],\n",
       " [(0, 5, 8)],\n",
       " [(0, 5, 9)],\n",
       " [(0, 5, 10)],\n",
       " [(0, 5, 11)],\n",
       " [(0, 5, 12)],\n",
       " [(0, 5, 13)],\n",
       " [(0, 5, 14)],\n",
       " [(0, 5, 15)],\n",
       " [(0, 6, 8)],\n",
       " [(0, 6, 9)],\n",
       " [(0, 6, 10), (0, 6, 11)],\n",
       " [(0, 6, 12)],\n",
       " [(0, 6, 13)],\n",
       " [(0, 6, 14)],\n",
       " [(0, 6, 15)],\n",
       " [(0, 7, 9)],\n",
       " [(0, 7, 13)],\n",
       " [(0, 8, 9)],\n",
       " [(0, 8, 10), (0, 8, 11)],\n",
       " [(0, 8, 12)],\n",
       " [(0, 8, 13)],\n",
       " [(0, 8, 14)],\n",
       " [(0, 8, 15)],\n",
       " [(0, 9, 10)],\n",
       " [(0, 9, 11)],\n",
       " [(0, 9, 12)],\n",
       " [(0, 9, 13)],\n",
       " [(0, 9, 14)],\n",
       " [(0, 9, 15)],\n",
       " [(0, 10, 12), (0, 11, 12)],\n",
       " [(0, 10, 13)],\n",
       " [(0, 10, 14), (0, 11, 14)],\n",
       " [(0, 10, 15), (0, 11, 15)],\n",
       " [(0, 11, 13)],\n",
       " [(0, 12, 13)],\n",
       " [(0, 12, 14)],\n",
       " [(0, 12, 15)],\n",
       " [(0, 13, 14)],\n",
       " [(0, 13, 15)],\n",
       " [(0, 14, 15)],\n",
       " [(1, 2, 3)],\n",
       " [(1, 2, 4)],\n",
       " [(1, 2, 5)],\n",
       " [(1, 2, 6), (1, 2, 9)],\n",
       " [(1, 2, 7), (1, 2, 13)],\n",
       " [(1, 2, 8)],\n",
       " [(1, 2, 10)],\n",
       " [(1, 2, 11)],\n",
       " [(1, 2, 12)],\n",
       " [(1, 2, 14)],\n",
       " [(1, 2, 15)],\n",
       " [(1, 3, 4)],\n",
       " [(1, 3, 5)],\n",
       " [(1, 3, 6), (1, 3, 9)],\n",
       " [(1, 3, 7), (1, 3, 13)],\n",
       " [(1, 3, 8)],\n",
       " [(1, 3, 10)],\n",
       " [(1, 3, 11)],\n",
       " [(1, 3, 12)],\n",
       " [(1, 3, 14)],\n",
       " [(1, 3, 15)],\n",
       " [(1, 4, 5)],\n",
       " [(1, 4, 6)],\n",
       " [(1, 4, 7)],\n",
       " [(1, 4, 8)],\n",
       " [(1, 4, 9)],\n",
       " [(1, 4, 10)],\n",
       " [(1, 4, 11)],\n",
       " [(1, 4, 12)],\n",
       " [(1, 4, 13)],\n",
       " [(1, 4, 14)],\n",
       " [(1, 4, 15)],\n",
       " [(1, 5, 6), (1, 5, 9)],\n",
       " [(1, 5, 7), (1, 5, 13)],\n",
       " [(1, 5, 8)],\n",
       " [(1, 5, 10)],\n",
       " [(1, 5, 11)],\n",
       " [(1, 5, 12)],\n",
       " [(1, 5, 14)],\n",
       " [(1, 5, 15)],\n",
       " [(1, 6, 7), (1, 9, 13)],\n",
       " [(1, 6, 8), (1, 8, 9)],\n",
       " [(1, 6, 10)],\n",
       " [(1, 6, 11), (1, 9, 11)],\n",
       " [(1, 6, 12)],\n",
       " [(1, 6, 13), (1, 7, 9)],\n",
       " [(1, 6, 14)],\n",
       " [(1, 6, 15)],\n",
       " [(1, 7, 8), (1, 8, 13)],\n",
       " [(1, 7, 10)],\n",
       " [(1, 7, 11), (1, 11, 13)],\n",
       " [(1, 7, 12)],\n",
       " [(1, 7, 14)],\n",
       " [(1, 7, 15)],\n",
       " [(1, 8, 10)],\n",
       " [(1, 8, 11)],\n",
       " [(1, 8, 12)],\n",
       " [(1, 8, 14)],\n",
       " [(1, 8, 15)],\n",
       " [(1, 9, 10)],\n",
       " [(1, 9, 12)],\n",
       " [(1, 9, 14)],\n",
       " [(1, 9, 15)],\n",
       " [(1, 10, 11)],\n",
       " [(1, 10, 12)],\n",
       " [(1, 10, 13)],\n",
       " [(1, 10, 14)],\n",
       " [(1, 10, 15)],\n",
       " [(1, 11, 12)],\n",
       " [(1, 11, 14)],\n",
       " [(1, 11, 15)],\n",
       " [(1, 12, 13)],\n",
       " [(1, 12, 14)],\n",
       " [(1, 12, 15)],\n",
       " [(1, 13, 14)],\n",
       " [(1, 13, 15)],\n",
       " [(1, 14, 15)],\n",
       " [(2, 3, 4)],\n",
       " [(2, 3, 5)],\n",
       " [(2, 3, 6), (2, 3, 9)],\n",
       " [(2, 3, 7), (2, 3, 13)],\n",
       " [(2, 3, 8)],\n",
       " [(2, 3, 10)],\n",
       " [(2, 3, 11)],\n",
       " [(2, 3, 12)],\n",
       " [(2, 3, 14)],\n",
       " [(2, 3, 15)],\n",
       " [(2, 4, 5), (2, 13, 14)],\n",
       " [(2, 4, 6), (2, 6, 7), (2, 9, 13)],\n",
       " [(2, 4, 8), (2, 7, 8), (2, 8, 13)],\n",
       " [(2, 4, 9)],\n",
       " [(2, 4, 10), (2, 7, 11), (2, 11, 13)],\n",
       " [(2, 4, 11), (2, 7, 10)],\n",
       " [(2, 4, 12)],\n",
       " [(2, 4, 13)],\n",
       " [(2, 4, 14), (2, 7, 14)],\n",
       " [(2, 4, 15), (2, 7, 15)],\n",
       " [(2, 5, 6), (2, 5, 9), (2, 9, 14)],\n",
       " [(2, 5, 7), (2, 5, 13)],\n",
       " [(2, 5, 8), (2, 8, 14)],\n",
       " [(2, 5, 10), (2, 10, 14), (2, 11, 14)],\n",
       " [(2, 5, 11)],\n",
       " [(2, 5, 12)],\n",
       " [(2, 5, 15)],\n",
       " [(2, 6, 8), (2, 8, 9)],\n",
       " [(2, 6, 10), (2, 6, 11), (2, 9, 11)],\n",
       " [(2, 6, 12)],\n",
       " [(2, 6, 13), (2, 7, 9)],\n",
       " [(2, 6, 14)],\n",
       " [(2, 6, 15)],\n",
       " [(2, 7, 12)],\n",
       " [(2, 8, 10), (2, 8, 11)],\n",
       " [(2, 8, 12)],\n",
       " [(2, 8, 15)],\n",
       " [(2, 9, 10)],\n",
       " [(2, 9, 12)],\n",
       " [(2, 9, 15)],\n",
       " [(2, 10, 12)],\n",
       " [(2, 10, 13)],\n",
       " [(2, 10, 15), (2, 11, 15)],\n",
       " [(2, 11, 12)],\n",
       " [(2, 12, 13)],\n",
       " [(2, 12, 14)],\n",
       " [(2, 12, 15)],\n",
       " [(2, 13, 15)],\n",
       " [(2, 14, 15)],\n",
       " [(3, 4, 5), (3, 12, 13)],\n",
       " [(3, 4, 6), (3, 6, 7), (3, 9, 13)],\n",
       " [(3, 4, 8), (3, 7, 8), (3, 8, 13)],\n",
       " [(3, 4, 9)],\n",
       " [(3, 4, 10), (3, 7, 11), (3, 11, 13)],\n",
       " [(3, 4, 11), (3, 7, 10)],\n",
       " [(3, 4, 12), (3, 7, 12)],\n",
       " [(3, 4, 13)],\n",
       " [(3, 4, 14)],\n",
       " [(3, 4, 15), (3, 7, 15)],\n",
       " [(3, 5, 6), (3, 5, 9), (3, 9, 12)],\n",
       " [(3, 5, 7), (3, 5, 13)],\n",
       " [(3, 5, 8), (3, 8, 12)],\n",
       " [(3, 5, 10), (3, 10, 12), (3, 11, 12)],\n",
       " [(3, 5, 11)],\n",
       " [(3, 5, 14)],\n",
       " [(3, 5, 15)],\n",
       " [(3, 6, 8), (3, 8, 9)],\n",
       " [(3, 6, 10), (3, 6, 11), (3, 9, 11)],\n",
       " [(3, 6, 12)],\n",
       " [(3, 6, 13), (3, 7, 9)],\n",
       " [(3, 6, 14)],\n",
       " [(3, 6, 15)],\n",
       " [(3, 7, 14)],\n",
       " [(3, 8, 10), (3, 8, 11)],\n",
       " [(3, 8, 14)],\n",
       " [(3, 8, 15)],\n",
       " [(3, 9, 10)],\n",
       " [(3, 9, 14)],\n",
       " [(3, 9, 15)],\n",
       " [(3, 10, 13)],\n",
       " [(3, 10, 14)],\n",
       " [(3, 10, 15), (3, 11, 15)],\n",
       " [(3, 11, 14)],\n",
       " [(3, 12, 14)],\n",
       " [(3, 12, 15)],\n",
       " [(3, 13, 14)],\n",
       " [(3, 13, 15)],\n",
       " [(3, 14, 15)],\n",
       " [(4, 5, 6), (9, 12, 13), (9, 13, 14)],\n",
       " [(4, 5, 7)],\n",
       " [(4, 5, 8), (8, 12, 13), (8, 13, 14)],\n",
       " [(4, 5, 9)],\n",
       " [(4, 5, 10), (11, 12, 13), (11, 13, 14)],\n",
       " [(4, 5, 11)],\n",
       " [(4, 5, 12)],\n",
       " [(4, 5, 13)],\n",
       " [(4, 5, 14)],\n",
       " [(4, 5, 15)],\n",
       " [(4, 6, 8), (6, 7, 8), (8, 9, 13)],\n",
       " [(4, 6, 9)],\n",
       " [(4, 6, 10), (6, 7, 11), (9, 11, 13)],\n",
       " [(4, 6, 11), (6, 7, 10)],\n",
       " [(4, 6, 12), (6, 7, 12)],\n",
       " [(4, 6, 13)],\n",
       " [(4, 6, 14), (6, 7, 14)],\n",
       " [(4, 6, 15), (6, 7, 15)],\n",
       " [(4, 7, 9)],\n",
       " [(4, 7, 10), (4, 7, 11)],\n",
       " [(4, 7, 13)],\n",
       " [(4, 8, 9)],\n",
       " [(4, 8, 10), (7, 8, 11), (8, 11, 13)],\n",
       " [(4, 8, 11), (7, 8, 10)],\n",
       " [(4, 8, 12), (7, 8, 12)],\n",
       " [(4, 8, 13)],\n",
       " [(4, 8, 14), (7, 8, 14)],\n",
       " [(4, 8, 15), (7, 8, 15)],\n",
       " [(4, 9, 10)],\n",
       " [(4, 9, 11)],\n",
       " [(4, 9, 12)],\n",
       " [(4, 9, 13)],\n",
       " [(4, 9, 14)],\n",
       " [(4, 9, 15)],\n",
       " [(4, 10, 11), (7, 10, 11)],\n",
       " [(4, 10, 12), (7, 11, 12)],\n",
       " [(4, 10, 13)],\n",
       " [(4, 10, 14), (7, 11, 14)],\n",
       " [(4, 10, 15), (7, 11, 15)],\n",
       " [(4, 11, 12), (7, 10, 12)],\n",
       " [(4, 11, 13)],\n",
       " [(4, 11, 14), (7, 10, 14)],\n",
       " [(4, 11, 15), (7, 10, 15)],\n",
       " [(4, 12, 13)],\n",
       " [(4, 12, 14)],\n",
       " [(4, 12, 15), (7, 12, 15)],\n",
       " [(4, 13, 14)],\n",
       " [(4, 13, 15)],\n",
       " [(4, 14, 15), (7, 14, 15)],\n",
       " [(5, 6, 7), (5, 9, 13)],\n",
       " [(5, 6, 8), (5, 8, 9), (8, 9, 12), (8, 9, 14)],\n",
       " [(5, 6, 10), (9, 11, 12), (9, 11, 14)],\n",
       " [(5, 6, 11), (5, 9, 11)],\n",
       " [(5, 6, 12)],\n",
       " [(5, 6, 13), (5, 7, 9)],\n",
       " [(5, 6, 14)],\n",
       " [(5, 6, 15)],\n",
       " [(5, 7, 8), (5, 8, 13)],\n",
       " [(5, 7, 10)],\n",
       " [(5, 7, 11), (5, 11, 13)],\n",
       " [(5, 7, 12)],\n",
       " [(5, 7, 14)],\n",
       " [(5, 7, 15)],\n",
       " [(5, 8, 10), (8, 10, 12), (8, 10, 14), (8, 11, 12), (8, 11, 14)],\n",
       " [(5, 8, 11)],\n",
       " [(5, 8, 15)],\n",
       " [(5, 9, 10), (9, 10, 12), (9, 10, 14)],\n",
       " [(5, 9, 15)],\n",
       " [(5, 10, 11)],\n",
       " [(5, 10, 13)],\n",
       " [(5, 10, 15)],\n",
       " [(5, 11, 12)],\n",
       " [(5, 11, 14)],\n",
       " [(5, 11, 15)],\n",
       " [(5, 12, 13)],\n",
       " [(5, 12, 14)],\n",
       " [(5, 12, 15)],\n",
       " [(5, 13, 14)],\n",
       " [(5, 13, 15)],\n",
       " [(5, 14, 15)],\n",
       " [(6, 7, 9), (6, 9, 13)],\n",
       " [(6, 7, 13), (7, 9, 13)],\n",
       " [(6, 8, 10), (6, 8, 11), (8, 9, 11)],\n",
       " [(6, 8, 12)],\n",
       " [(6, 8, 13), (7, 8, 9)],\n",
       " [(6, 8, 14)],\n",
       " [(6, 8, 15)],\n",
       " [(6, 9, 10)],\n",
       " [(6, 9, 12)],\n",
       " [(6, 9, 14)],\n",
       " [(6, 9, 15)],\n",
       " [(6, 10, 12), (6, 11, 12)],\n",
       " [(6, 10, 13)],\n",
       " [(6, 10, 14), (6, 11, 14)],\n",
       " [(6, 10, 15), (6, 11, 15)],\n",
       " [(6, 11, 13), (7, 9, 11)],\n",
       " [(6, 12, 13)],\n",
       " [(6, 12, 14)],\n",
       " [(6, 12, 15)],\n",
       " [(6, 13, 14)],\n",
       " [(6, 13, 15)],\n",
       " [(6, 14, 15)],\n",
       " [(7, 9, 10)],\n",
       " [(7, 9, 12)],\n",
       " [(7, 9, 14)],\n",
       " [(7, 9, 15)],\n",
       " [(7, 10, 13)],\n",
       " [(7, 12, 13)],\n",
       " [(7, 12, 14)],\n",
       " [(7, 13, 14)],\n",
       " [(7, 13, 15)],\n",
       " [(8, 9, 10)],\n",
       " [(8, 9, 15)],\n",
       " [(8, 10, 13)],\n",
       " [(8, 10, 15), (8, 11, 15)],\n",
       " [(8, 12, 15), (8, 14, 15)],\n",
       " [(8, 13, 15)],\n",
       " [(9, 10, 11)],\n",
       " [(9, 10, 13)],\n",
       " [(9, 10, 15)],\n",
       " [(9, 11, 15)],\n",
       " [(9, 12, 15), (9, 14, 15)],\n",
       " [(9, 13, 15)],\n",
       " [(10, 11, 13)],\n",
       " [(10, 12, 13), (10, 13, 14)],\n",
       " [(10, 12, 15), (10, 14, 15), (11, 12, 15), (11, 14, 15)],\n",
       " [(10, 13, 15)],\n",
       " [(11, 13, 15)],\n",
       " [(12, 13, 15), (13, 14, 15)]]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "two_faces=glued_faces(2); two_faces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "Cell \u001b[0;32mIn [47], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m three_faces\u001b[38;5;241m=\u001b[39mglued_faces(Integer(\u001b[38;5;241m3\u001b[39m)); three_faces\n",
      "Cell \u001b[0;32mIn [35], line 8\u001b[0m, in \u001b[0;36mglued_faces\u001b[0;34m(k)\u001b[0m\n\u001b[1;32m      6\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m equiv_class \u001b[38;5;129;01min\u001b[39;00m glued_verts:\n\u001b[1;32m      7\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28mlen\u001b[39m(equiv_class) \u001b[38;5;241m>\u001b[39m\u001b[38;5;241m=\u001b[39m Integer(\u001b[38;5;241m1\u001b[39m):\n\u001b[0;32m----> 8\u001b[0m         \u001b[38;5;28;01mif\u001b[39;00m \u001b[43mfaces_glued\u001b[49m\u001b[43m(\u001b[49m\u001b[43mi\u001b[49m\u001b[43m,\u001b[49m\u001b[43mequiv_class\u001b[49m\u001b[43m[\u001b[49m\u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\u001b[43mG\u001b[49m\u001b[43m,\u001b[49m\u001b[43mvertices\u001b[49m\u001b[43m)\u001b[49m:\n\u001b[1;32m      9\u001b[0m             equiv_class\u001b[38;5;241m.\u001b[39mappend(i)\n\u001b[1;32m     10\u001b[0m             found\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mTrue\u001b[39;00m\n",
      "Cell \u001b[0;32mIn [31], line 5\u001b[0m, in \u001b[0;36mfaces_glued\u001b[0;34m(face1, face2, G, vertices)\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mfaces_glued\u001b[39m(face1,face2,G,vertices):\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;28;01mfor\u001b[39;00m g \u001b[38;5;129;01min\u001b[39;00m G:\n\u001b[0;32m----> 5\u001b[0m         g_face1 \u001b[38;5;241m=\u001b[39m \u001b[43mact_face\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m,\u001b[49m\u001b[43mface1\u001b[49m\u001b[43m,\u001b[49m\u001b[43mvertices\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m      6\u001b[0m         \u001b[38;5;28;01mif\u001b[39;00m g_face1 \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m \u001b[38;5;129;01mand\u001b[39;00m \u001b[38;5;28mset\u001b[39m(g_face1) \u001b[38;5;241m==\u001b[39m \u001b[38;5;28mset\u001b[39m(face2):\n\u001b[1;32m      7\u001b[0m             \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m\n",
      "Cell \u001b[0;32mIn [44], line 5\u001b[0m, in \u001b[0;36mact_face\u001b[0;34m(g, face, vertices)\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mact_face\u001b[39m(g,face,vertices):\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m----> 5\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43m[\u001b[49m\u001b[43mvertices\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mindex\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mtuple\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mact_vect\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m,\u001b[49m\u001b[43mvertices\u001b[49m\u001b[43m[\u001b[49m\u001b[43mi\u001b[49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mfor\u001b[39;49;00m\u001b[43m \u001b[49m\u001b[43mi\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;129;43;01min\u001b[39;49;00m\u001b[43m \u001b[49m\u001b[43mface\u001b[49m\u001b[43m]\u001b[49m\n\u001b[1;32m      6\u001b[0m     \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m:\n\u001b[1;32m      7\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m\n",
      "Cell \u001b[0;32mIn [44], line 5\u001b[0m, in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mact_face\u001b[39m(g,face,vertices):\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m----> 5\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m [vertices\u001b[38;5;241m.\u001b[39mindex(\u001b[38;5;28mtuple\u001b[39m(\u001b[43mact_vect\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m,\u001b[49m\u001b[43mvertices\u001b[49m\u001b[43m[\u001b[49m\u001b[43mi\u001b[49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m)) \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m face]\n\u001b[1;32m      6\u001b[0m     \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m:\n\u001b[1;32m      7\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m\n",
      "Cell \u001b[0;32mIn [20], line 8\u001b[0m, in \u001b[0;36mact_vect\u001b[0;34m(g, v, basis)\u001b[0m\n\u001b[0;32m----> 8\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mact_vect\u001b[39m(g,v,basis\u001b[38;5;241m=\u001b[39midentity_matrix(N)): \u001b[38;5;28;01mreturn\u001b[39;00m (\u001b[43mbasis\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43minverse\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mg\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmatrix\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[38;5;241m*\u001b[39mbasis)\u001b[38;5;241m*\u001b[39mvector(v)\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/element.pyx:3939\u001b[0m, in \u001b[0;36msage.structure.element.Matrix.__mul__ (build/cythonized/sage/structure/element.c:24277)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   3937\u001b[0m \n\u001b[1;32m   3938\u001b[0m         if BOTH_ARE_ELEMENT(cl):\n\u001b[0;32m-> 3939\u001b[0m             return coercion_model.bin_op(left, right, mul)\n\u001b[1;32m   3940\u001b[0m \n\u001b[1;32m   3941\u001b[0m         cdef long value\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/coerce.pyx:1215\u001b[0m, in \u001b[0;36msage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:10711)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1213\u001b[0m if action is not None:\n\u001b[1;32m   1214\u001b[0m     if (<Action>action)._is_left:\n\u001b[0;32m-> 1215\u001b[0m         return (<Action>action)._act_(x, y)\n\u001b[1;32m   1216\u001b[0m     else:\n\u001b[1;32m   1217\u001b[0m         return (<Action>action)._act_(y, x)\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/matrix/action.pyx:265\u001b[0m, in \u001b[0;36msage.matrix.action.MatrixMatrixAction._act_ (build/cythonized/sage/matrix/action.c:5162)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    263\u001b[0m     A = A.change_ring(self._codomain._base)\n\u001b[1;32m    264\u001b[0m if B._parent._base is not self._codomain._base:\n\u001b[0;32m--> 265\u001b[0m     B = B.change_ring(self._codomain._base)\n\u001b[1;32m    266\u001b[0m if self.fix_sparseness:\n\u001b[1;32m    267\u001b[0m     if B.is_sparse_c():\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/matrix/matrix_sparse.pyx:73\u001b[0m, in \u001b[0;36msage.matrix.matrix_sparse.Matrix_sparse.change_ring (build/cythonized/sage/matrix/matrix_sparse.c:3442)\u001b[0;34m()\u001b[0m\n\u001b[1;32m     71\u001b[0m \n\u001b[1;32m     72\u001b[0m         M = sage.matrix.matrix_space.MatrixSpace(ring, self._nrows, self._ncols, sparse=self.is_sparse_c())\n\u001b[0;32m---> 73\u001b[0m         mat = M(self.dict(), coerce=True, copy=False)\n\u001b[1;32m     74\u001b[0m         if self._subdivisions is not None:\n\u001b[1;32m     75\u001b[0m             mat.subdivide(self.subdivisions())\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/parent.pyx:903\u001b[0m, in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9477)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    901\u001b[0m         return mor._call_(x)\n\u001b[1;32m    902\u001b[0m     else:\n\u001b[0;32m--> 903\u001b[0m         return mor._call_with_args(x, args, kwds)\n\u001b[1;32m    904\u001b[0m \n\u001b[1;32m    905\u001b[0m raise TypeError(_LazyString(\"No conversion defined from %s to %s\", (R, self), {}))\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/coerce_maps.pyx:172\u001b[0m, in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_with_args (build/cythonized/sage/structure/coerce_maps.c:5159)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    170\u001b[0m         return C._element_constructor(x)\n\u001b[1;32m    171\u001b[0m     else:\n\u001b[0;32m--> 172\u001b[0m         return C._element_constructor(x, **kwds)\n\u001b[1;32m    173\u001b[0m else:\n\u001b[1;32m    174\u001b[0m     if len(kwds) == 0:\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/matrix/matrix_space.py:945\u001b[0m, in \u001b[0;36mMatrixSpace._element_constructor_\u001b[0;34m(self, entries, **kwds)\u001b[0m\n\u001b[1;32m    835\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m_element_constructor_\u001b[39m(\u001b[38;5;28mself\u001b[39m, entries, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwds):\n\u001b[1;32m    836\u001b[0m     \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m    837\u001b[0m \u001b[38;5;124;03m    Construct an element of ``self`` from ``entries``.\u001b[39;00m\n\u001b[1;32m    838\u001b[0m \n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m    943\u001b[0m \u001b[38;5;124;03m        False\u001b[39;00m\n\u001b[1;32m    944\u001b[0m \u001b[38;5;124;03m    \"\"\"\u001b[39;00m\n\u001b[0;32m--> 945\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43melement_class\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mentries\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwds\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/matrix/matrix_rational_sparse.pyx:90\u001b[0m, in \u001b[0;36msage.matrix.matrix_rational_sparse.Matrix_rational_sparse.__init__ (build/cythonized/sage/matrix/matrix_rational_sparse.c:4082)\u001b[0;34m()\u001b[0m\n\u001b[1;32m     88\u001b[0m ma = MatrixArgs_init(parent, entries)\n\u001b[1;32m     89\u001b[0m cdef Rational z\n\u001b[0;32m---> 90\u001b[0m for t in ma.iter(coerce, True):\n\u001b[1;32m     91\u001b[0m     se = <SparseEntry>t\n\u001b[1;32m     92\u001b[0m     z = <Rational>se.entry\n",
      "File \u001b[0;32m/private/var/tmp/sage-10.1-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/matrix/args.pyx:542\u001b[0m, in \u001b[0;36miter (build/cythonized/sage/matrix/args.c:7298)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    540\u001b[0m if sparse:\n\u001b[1;32m    541\u001b[0m     for t in self.entries:\n\u001b[0;32m--> 542\u001b[0m         sig_check()\n\u001b[1;32m    543\u001b[0m         se = <SparseEntry>t\n\u001b[1;32m    544\u001b[0m         if convert and self.need_to_convert(se.entry):\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "three_faces=glued_faces(3); three_faces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "glued_face_list=[zero_faces,one_faces,two_faces]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10 x 70 dense matrix over Integer Ring (use the '.str()' method to see the entries)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1=boundary_map(1, glued_face_list, vertices); d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "70 x 381 dense matrix over Integer Ring (use the '.str()' method to see the entries)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2=boundary_map(2, glued_face_list, vertices); d2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "ename": "IndexError",
     "evalue": "list index out of range",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mIndexError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[0;32mIn [60], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m d3\u001b[38;5;241m=\u001b[39mboundary_map(Integer(\u001b[38;5;241m3\u001b[39m), glued_face_list, vertices); d3\n",
      "Cell \u001b[0;32mIn [55], line 2\u001b[0m, in \u001b[0;36mboundary_map\u001b[0;34m(k, glued_face_list, vertices)\u001b[0m\n\u001b[0;32m----> 2\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mboundary_map\u001b[39m(k,glued_face_list,vertices): \u001b[38;5;28;01mreturn\u001b[39;00m matrix(ZZ,[boundary(face[Integer(\u001b[38;5;241m0\u001b[39m)],glued_face_list,vertices) \u001b[38;5;28;01mfor\u001b[39;00m face \u001b[38;5;129;01min\u001b[39;00m \u001b[43mglued_face_list\u001b[49m\u001b[43m[\u001b[49m\u001b[43mk\u001b[49m\u001b[43m]\u001b[49m])\u001b[38;5;241m.\u001b[39mtranspose()\n",
      "\u001b[0;31mIndexError\u001b[0m: list index out of range"
     ]
    }
   ],
   "source": [
    "d3=boundary_map(3, glued_face_list, vertices); d3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "#form the chain complex with truncation at degree 2\n",
    "cc = chain_complex(max_degree=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: Z, 1: 0, 2: Z^320}"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#compute the homology of the chain complex\n",
    "cc.homology()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10, 70, 381]"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(map(lambda x: len(x), glued_face_list))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "interpreter": {
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  "kernelspec": {
   "display_name": "SageMath 10.1",
   "language": "sage",
   "name": "sagemath"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
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   "file_extension": ".py",
   "mimetype": "text/x-python",
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