{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Knot theory in Sage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Representation of knots.\n",
    "\n",
    "A Knot (or a Link) can be represented by three different conventions:\n",
    "\n",
    "- Oriented Gauss code\n",
    "- PlanarDiagram code\n",
    "- Braid\n",
    "\n",
    "\n",
    "Notice that these representations are in fact representations of a particular diagram."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oriented Gauss code:\n",
    "\n",
    "It is a list of two lists. The first one contains a list with the crossings of each component. The second list contains the orientation of each crossing.\n",
    "\n",
    "- Number the crossings from 1 to n\n",
    "- For each component, list the crossings it goes through; with a positive sign if it goes over the crossing and negative if it goes under.\n",
    "- The second list contains 1's and -1's indicating the orientation of each crossing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Planar Diagram Code\n",
    "\n",
    "- Number the strands (beware, overcrossings also separate strands).\n",
    "- Orient the Knot/Link\n",
    "- Represent each crossing with a list with the four strands, starting by the incomimg undercrossing and going clockwise\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Braid \n",
    "\n",
    "Give a braid (usually generated by the Tietze list). The knot/link is created as the closure of the braid."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise\n",
    "\n",
    "Create the figure eight knot with each convention."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K = Knot([[[1,-2,3,-1,2,-3]],[1,1,1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Knot represented by 3 crossings"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K.alexander_polynomial??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K2 = Knot([[4,2,1,6],[2,3,5,1],[3,4,6,5]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Knot represented by 3 crossings"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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5nAUkltcAIDFYZgMSz1lAYnkNABKDZTYg8ZwFJJbXACAxWGYDEs9JQNq2jeU1AEgkltmA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\nGP8fwxtssF5DXpQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "Graphics object consisting of 14 graphics primitives"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[1, -2, 3, -1, 2, -3]], [1, 1, 1]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.oriented_gauss_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[4, 1, 5, 2], [2, 5, 3, 6], [6, 3, 1, 4]]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.pd_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[2, -3, 1, -2, 3, -1]], [-1, -1, -1]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K2.oriented_gauss_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[4, 2, 1, 6], [2, 3, 5, 1], [3, 4, 6, 5]]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K2.pd_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "B = BraidGroup(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "b = B([1,1,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K3 = Knot(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Knot represented by 3 crossings"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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tu48LCFfTpraStHChNHCglJvruqLYIiCVwcsvS+PHW89R06auqwGAg+OIP0qr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\nAhIAAEAIAhIAAECI/wf4ENB2P7TkDgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 14 graphics primitives"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K3.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+lNABaf78wi7ZnFWDaGjc2BpF/vqr60oAoHhHHSU9/rj09NO2eBslk7ABaccO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\ngg==\n",
      "text/plain": [
       "Graphics object consisting of 14 graphics primitives"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K2.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Represent knots and links:\n",
    "\n",
    "- The .plot method creates a picture of the knot/link\n",
    "\n",
    "Construct and plot the two trefoil knots, the Hopf link with two components, and the figure eight."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K8=Knot([[[1,-2,4,-3,2,-1,3,-4]],[1,1,-1,-1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4vCZNvF3qT0AC4Bt5eSzxB1A00aX+69Z58/oEJAC+sW4dS/wBFE30PuHVPCQCEgDfYIk/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\nSAAAmS/WqyR73HooSgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 22 graphics primitives"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8UFq+3HU1+aIekCSm2fDnFi6URo2y0aNixVxXAwDwUu/eUoUK0pAhrivJ5yQg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\n4cL8wDR7tp0VFz4brlIlG2063FWtmlS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      "text/plain": [
       "Graphics object consisting of 21 graphics primitives"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.braid().plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Construction of knots and links from others:\n",
    "\n",
    "Look for the help of the .mirror_image and .connected_sum methods. Create new knots from the previous ones.\n",
    "\n",
    "Notice that .connected_sum does not work on Links. Why?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Knot represented by 4 crossings"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.mirror_image()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Invariants\n",
    "\n",
    "Go through the methods available for knots and links and identify those that give invariants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-t^-1 + 3 - t"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.alexander_polynomial()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K8.alexander_polynomial??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t^2 - t - 1/t + 1/t^2 + 1"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.jones_polynomial()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 7, -4], [2, -5, -7], [3, -8, 5], [4, 8], [6, -1, -3], [-2, -6]]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.regions()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[6, 1, 7, 2], [2, 5, 3, 6], [4, 8, 5, 7], [8, 4, 1, 3]]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.pd_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, -1, -1]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 7, 5, 3], [2, 6], [4, 8]]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K8.seifert_circles()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K8.jones_polynomial??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Joint project:\n",
    "\n",
    "Write a function that, given a knot or link, returns the fundamental group of its complement.\n",
    "\n",
    "\n",
    "#### Help:\n",
    "\n",
    "Finitely presented groups can be generated in Sage as quotients of Free Groups.\n",
    "\n",
    "FreeGroups are generated passing the number of generators. Their n'th generator can be retrieved by F.gen(n).\n",
    "\n",
    "#### Auxiliary tasks:\n",
    "\n",
    "Write a function that identifies the "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F= FreeGroup(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Free Group on generators {x0, x1, x2}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Defining x0, x1, x2\n"
     ]
    }
   ],
   "source": [
    "F.inject_variables()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x0*x2^2*x1^-1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0*x2^2/x1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G = F.quotient([x0^2,x1^2,x2^2,x1*x0/x1/x0,x1*x2/x1/x2,x0*x2/x0/x2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finitely presented group < x0, x1, x2 | x0^2, x1^2, x2^2, x1*x0*x1^-1*x0^-1, x1*x2*x1^-1*x2^-1, x0*x2*x0^-1*x2^-1 >"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.cardinality()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Permutation Group with generators [(1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,7)(6,8), (1,4)(2,6)(3,7)(5,8)]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.as_permutation_group()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G.as_permutation_group??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x1*x2^-1*x1^-1*x0*x1*x2*x1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-1*x0^-1,\n",
       " x1*x2^-1*x1^-1*x0*x1*x2*x1^-1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-1*x0*x1*x2*x1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-2,\n",
       " x1*x2^-1*x1^-1*x0*x1*x2*x1^-1*x2^-1]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[i*K8.braid()*i^-1 for i in FreeGroup(K8.braid().strands()).gens()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fundamental_group(K):\n",
    "    gauss_code = K.oriented_gauss_code()\n",
    "    crossing_list=gauss_code[0][0]\n",
    "    negative_indices = [i for i in range(len(crossing_list)) if crossing_list[i]<0]\n",
    "    gens = [crossing_list[negative_indices[i]:negative_indices[i+1]+1] for i in range(len(negative_indices)-1)]\n",
    "    gens.append(crossing_list[negative_indices[-1]:]+crossing_list[:negative_indices[0]+1])\n",
    "    relations = []\n",
    "    for cros in range(len(gauss_code[1])):\n",
    "        gi = [g for g in gens if cros+1 in g][0]\n",
    "        gj = [g for g in gens if -(cros+1)==g[-1]][0]\n",
    "        giind = gens.index(gi)\n",
    "        gjind = gens.index(gj)\n",
    "        gkind = gjind + 1\n",
    "        if gkind >= len(gens):\n",
    "            gkind = 0\n",
    "        if gauss_code[1][cros] > 0:\n",
    "            relations.append([giind+1,gkind+1,-giind-1,-gjind-1])\n",
    "        else:\n",
    "            relations.append([giind+1,gjind+1,-giind-1,-gkind-1])\n",
    "    F = FreeGroup(len(gens))\n",
    "    return F.quotient(relations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "G = fundamental_group(K8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finitely presented group < x0, x3 | x3*x0^-1*x3*x0*x3^-1*x0^-1*x3*x0^-1*x3^-1*x0 >"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.simplified()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[1, -2, 4, -3, 2, -1, 3, -4]], [1, 1, -1, -1]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gauss_code=K8.oriented_gauss_code()\n",
    "gauss_code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, -2, 4, -3, 2, -1, 3, -4]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "crossing_list=gauss_code[0][0]\n",
    "crossing_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 3, 5, 7]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "negative_indices = [i for i in range(len(crossing_list)) if crossing_list[i]<0]\n",
    "negative_indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[-2, 4, -3], [-3, 2, -1], [-1, 3, -4]]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gens = [crossing_list[negative_indices[i]:negative_indices[i+1]+1] for i in range(len(negative_indices)-1)]\n",
    "gens"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "gens.append(crossing_list[negative_indices[-1]:]+crossing_list[:negative_indices[0]+1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[-2, 4, -3], [-3, 2, -1], [-1, 3, -4], [-4, 1, -2]]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gens"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cros = 0\n",
    "cros+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-4, 1, -2]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gi = [g for g in gens if cros+1 in g][0]\n",
    "gi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-3, 2, -1]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gj = [g for g in gens if -(cros+1)==g[-1]][0]\n",
    "gj"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "giind = gens.index(gi)\n",
    "giind"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gjind = gens.index(gj)\n",
    "gjind"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "B = BraidGroup(2)\n",
    "K = Knot(B([1,1,1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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tu48LCFfTpraStHChNHCglJvruqLYIiCVwcsvS+PHW89R06auqwGAg+OIP0qr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\nAhIAAEAIAhIAAECI/wf4ENB2P7TkDgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 14 graphics primitives"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G = fundamental_group(K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finitely presented group < x0, x1, x2 | x2*x1*x2^-1*x0^-1, x1*x0*x1^-1*x2^-1, x0*x2*x0^-1*x1^-1 >"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Knot represented by 3 crossings"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[1, -2, 3, -1, 2, -3]], [1, 1, 1]]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.oriented_gauss_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[2, 1, 3, 4], [4, 3, 5, 6], [6, 5, 1, 2]]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.pd_code()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "s^3"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.braid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finitely presented group < x0, x1 | x0^-1*x1*x0*x1*x0^-1*x1^-1 >"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.simplified()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K5 = Knot(B([1,1,1,1,1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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rJNHlaUFieg0AgGDJypKqVZOOOcZ1kujyrCAxvQYAQPBkZUknnywlJ7tOEl2e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\nQmCC\n",
      "text/plain": [
       "Graphics object consisting of 20 graphics primitives"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K5.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G5 = fundamental_group(K5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finitely presented group < x0, x1, x2, x3, x4 | x4*x2*x4^-1*x1^-1, x2*x0*x2^-1*x4^-1, x0*x3*x0^-1*x2^-1, x3*x1*x3^-1*x0^-1, x1*x4*x1^-1*x3^-1 >"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finitely presented group < x1, x4 | x4^-1*x1^-1*(x4*x1)^2*x4*x1^-1*x4^-1*x1^-1 >"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G5.simplified()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "R = G5.rewriting_system()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-49-2a567dd874d9>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mR\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmake_confluent\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m/home/mmarco/sagedays83/sage-7.3/local/lib/python2.7/site-packages/sage/groups/finitely_presented.pyc\u001b[0m in \u001b[0;36mmake_confluent\u001b[1;34m(self)\u001b[0m\n\u001b[0;32m    701\u001b[0m         \"\"\"\n\u001b[0;32m    702\u001b[0m         \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 703\u001b[1;33m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_gap\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mMakeConfluent\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    704\u001b[0m         \u001b[1;32mexcept\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    705\u001b[0m             \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'could not make the system confluent'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/mmarco/sagedays83/sage-7.3/src/sage/libs/gap/element.pyx\u001b[0m in \u001b[0;36msage.libs.gap.element.GapElement_MethodProxy.__call__ (/home/mmarco/sagedays83/sage-7.3/src/build/cythonized/sage/libs/gap/element.c:16776)\u001b[1;34m()\u001b[0m\n\u001b[0;32m   2165\u001b[0m             \u001b[1;32mreturn\u001b[0m \u001b[0mGapElement_Function\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m__call__\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m*\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfirst_argument\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mlist\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2166\u001b[0m         \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2167\u001b[1;33m             \u001b[1;32mreturn\u001b[0m \u001b[0mGapElement_Function\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m__call__\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfirst_argument\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2168\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2169\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/mmarco/sagedays83/sage-7.3/src/sage/libs/gap/element.pyx\u001b[0m in \u001b[0;36msage.libs.gap.element.GapElement_Function.__call__ (/home/mmarco/sagedays83/sage-7.3/src/build/cythonized/sage/libs/gap/element.c:15606)\u001b[1;34m()\u001b[0m\n\u001b[0;32m   2006\u001b[0m         \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2007\u001b[0m             \u001b[0mlibgap_enter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2008\u001b[1;33m             \u001b[0msig_on\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2009\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mn\u001b[0m \u001b[1;33m==\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2010\u001b[0m                 \u001b[0mresult\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mlibGAP_CALL_0ARGS\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mvalue\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32msrc/cysignals/signals.pyx\u001b[0m in \u001b[0;36mcysignals.signals.sig_raise_exception (build/src/cysignals/signals.c:1125)\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "R.make_confluent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Rewriting system of Finitely presented group < x0, x1, x2, x3, x4 | x4*x2*x4^-1*x1^-1, x2*x0*x2^-1*x4^-1, x0*x3*x0^-1*x2^-1, x3*x1*x3^-1*x0^-1, x1*x4*x1^-1*x3^-1 >\n",
       "with rules:\n",
       "    x1^-1*x3^-1    --->    x0^-1*x2^-1\n",
       "    x1*x4    --->    x0*x3\n",
       "    x2^-1*x0    --->    x0*x3^-1\n",
       "    x2^-1*x1    --->    x0*x4^-1\n",
       "    x2^-1*x3    --->    x0*x1^-1\n",
       "    x2^-1*x4^-1    --->    x0^-1*x2^-1\n",
       "    x2^-1*x4    --->    x0*x2^-1\n",
       "    x2*x0    --->    x0*x3\n",
       "    x3^-1*x0^-1    --->    x0^-1*x2^-1\n",
       "    x3^-1*x0    --->    x1*x3^-1\n",
       "    x3^-1*x1    --->    x1*x4^-1\n",
       "    x3^-1*x2    --->    x1*x0^-1\n",
       "    x3^-1*x4    --->    x1*x2^-1\n",
       "    x3*x0^-1    --->    x0^-1*x2\n",
       "    x3*x1^-1    --->    x0^-1*x3\n",
       "    x3*x1    --->    x0*x3\n",
       "    x3*x2^-1    --->    x0^-1*x4\n",
       "    x3*x4^-1    --->    x0^-1*x1\n",
       "    x4^-1*x0    --->    x2*x3^-1\n",
       "    x4^-1*x1^-1    --->    x0^-1*x2^-1\n",
       "    x4^-1*x1    --->    x2*x4^-1\n",
       "    x4^-1*x2    --->    x2*x0^-1\n",
       "    x4^-1*x3    --->    x2*x1^-1\n",
       "    x4*x0^-1    --->    x1^-1*x2\n",
       "    x4*x1^-1    --->    x1^-1*x3\n",
       "    x4*x2^-1    --->    x1^-1*x4\n",
       "    x4*x2    --->    x0*x3\n",
       "    x4*x3^-1    --->    x1^-1*x0\n",
       "    x1^-1*x0^-1*x2^-1    --->    x0^-1*x2^-1*x0^-1\n",
       "    x1^-1*x0*x3    --->    x4\n",
       "    x1*x0^-1*x2^-1    --->    x3^-1\n",
       "    x1*x0*x3    --->    x0*x3*x2\n",
       "    x2^-1*x0^-1*x1    --->    x0*x1^-1*x4^-1\n",
       "    x2^-1*x0^-1*x2^-1    --->    x0^-1*x2^-1*x1^-1\n",
       "    x2^-1*x0^-1*x2    --->    x0*x1^-1*x0^-1\n",
       "    x2^-1*x0^-1*x3    --->    x0*x1^-2\n",
       "    x2^-1*x0^-1*x4    --->    x0*x1^-1*x2^-1\n",
       "    x2^-1*x1^-1*x0    --->    x0*x2^-1*x3^-1\n",
       "    x2^-1*x1^-1*x2    --->    x0*x2^-1*x0^-1\n",
       "    x2^-1*x1^-1*x3    --->    x0*x2^-1*x1^-1\n",
       "    x2^-1*x1^-1*x4    --->    x0*x2^-2\n",
       "    x2*x0^-1*x2^-1    --->    x4^-1\n",
       "    x3^-1*x1^-1*x0    --->    x1*x2^-1*x3^-1\n",
       "    x3^-1*x1^-1*x2    --->    x1*x2^-1*x0^-1\n",
       "    x3^-1*x1^-1*x3    --->    x1*x2^-1*x1^-1\n",
       "    x3^-1*x1^-1*x4    --->    x1*x2^-2\n",
       "    x3*x0*x1^-1    --->    x0^-1*x4*x3\n",
       "    x3*x0*x2^-1    --->    x0^-1*x4^2\n",
       "    x3*x0*x3^-1    --->    x0^-1*x4*x0\n",
       "    x3*x0*x3    --->    x0*x3*x4\n",
       "    x3*x0*x4^-1    --->    x0^-1*x4*x1\n",
       "    x3*x2*x0^-1    --->    x0^-1*x1*x2\n",
       "    x3*x2*x1^-1    --->    x0^-1*x1*x3\n",
       "    x3*x2*x3^-1    --->    x0^-1*x1*x0\n",
       "    x3*x2*x4^-1    --->    x0^-1*x1^2\n",
       "    x4^-1*x0^-1*x2^-1    --->    x0^-1*x2^-1*x3^-1\n",
       "    x4^-1*x0^-1*x2    --->    x2*x1^-1*x0^-1\n",
       "    x4^-1*x0^-1*x3    --->    x2*x1^-2\n",
       "    x4^-1*x0^-1*x4    --->    x2*x1^-1*x2^-1\n",
       "    x4*x0*x1^-1    --->    x1^-1*x4*x3\n",
       "    x4*x0*x2^-1    --->    x1^-1*x4^2\n",
       "    x4*x0*x3^-1    --->    x1^-1*x4*x0\n",
       "    x4*x0*x3    --->    x0*x3*x0\n",
       "    x4*x0*x4^-1    --->    x1^-1*x4*x1\n",
       "    x4*x1*x0^-1    --->    x1^-1*x0*x2\n",
       "    x4*x1*x2^-1    --->    x1^-1*x0*x4\n",
       "    x4*x1*x3^-1    --->    x1^-1*x0^2\n",
       "    x4*x1*x4^-1    --->    x1^-1*x0*x1\n",
       "    x1^-1*x0^-2*x2^-1    --->    x0^-1*x2^-1*x0^-1*x4^-1\n",
       "    x1^-1*x0^2*x3    --->    x4*x1\n",
       "    x1*x0^-2*x2^-1    --->    x3^-1*x4^-1\n",
       "    x1*x0^2*x3    --->    x0*x3*x2*x1\n",
       "    x2^-1*x0^-2*x2^-1    --->    x0^-1*x2^-1*x1^-1*x4^-1\n",
       "    x2^-1*x0^-1*x1^-1*x0    --->    x0*x1^-1*x2^-1*x3^-1\n",
       "    x2^-1*x0^-1*x1^-1*x2    --->    x0*x1^-1*x2^-1*x0^-1\n",
       "    x2^-1*x0^-1*x1^-1*x3    --->    x0*x1^-1*x2^-1*x1^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1    --->    x0*x2^-1*x1^-1*x4^-1\n",
       "    x2^-1*x1^-1*x0^-1*x2    --->    x0*x2^-1*x1^-1*x0^-1\n",
       "    x2^-1*x1^-1*x0^-1*x3    --->    x0*x2^-1*x1^-2\n",
       "    x2^-1*x1^-1*x0^-1*x4    --->    x0*x2^-1*x1^-1*x2^-1\n",
       "    x2^-1*x1^-2*x0    --->    x0*x2^-2*x3^-1\n",
       "    x2^-1*x1^-2*x2    --->    x0*x2^-2*x0^-1\n",
       "    x2^-1*x1^-2*x3    --->    x0*x2^-2*x1^-1\n",
       "    x2^-1*x1^-2*x4    --->    x0*x2^-3\n",
       "    x2*x0^-2*x2^-1    --->    x4^-2\n",
       "    x3^-1*x1^-1*x0^-1*x1    --->    x1*x2^-1*x1^-1*x4^-1\n",
       "    x3^-1*x1^-1*x0^-1*x2    --->    x1*x2^-1*x1^-1*x0^-1\n",
       "    x3^-1*x1^-1*x0^-1*x3    --->    x1*x2^-1*x1^-2\n",
       "    x3^-1*x1^-1*x0^-1*x4    --->    x1*x2^-1*x1^-1*x2^-1\n",
       "    x3^-1*x1^-2*x0    --->    x1*x2^-2*x3^-1\n",
       "    x3^-1*x1^-2*x2    --->    x1*x2^-2*x0^-1\n",
       "    x3^-1*x1^-2*x3    --->    x1*x2^-2*x1^-1\n",
       "    x3^-1*x1^-2*x4    --->    x1*x2^-3\n",
       "    x3*x0^2*x1^-1    --->    x0^-1*x4^2*x3\n",
       "    x3*x0^2*x2^-1    --->    x0^-1*x4^3\n",
       "    x3*x0^2*x3^-1    --->    x0^-1*x4^2*x0\n",
       "    x3*x0^2*x3    --->    x0*x3*x4*x1\n",
       "    x3*x0^2*x4^-1    --->    x0^-1*x4^2*x1\n",
       "    x3*x0*x1*x0^-1    --->    x0^-1*x4*x0*x2\n",
       "    x3*x0*x1*x2^-1    --->    x0^-1*x4*x0*x4\n",
       "    x3*x0*x1*x3^-1    --->    x0^-1*x4*x0^2\n",
       "    x3*x0*x1*x4^-1    --->    x0^-1*x4*x0*x1\n",
       "    x3*x0*x2*x0^-1    --->    x0^-1*x4*x1*x2\n",
       "    x3*x0*x2*x3^-1    --->    x0^-1*x4*x1*x0\n",
       "    x3*x0*x2*x4^-1    --->    x0^-1*x4*x1^2\n",
       "    x3*x2*x1*x0^-1    --->    x0^-1*x1*x0*x2\n",
       "    x3*x2*x1*x2^-1    --->    x0^-1*x1*x0*x4\n",
       "    x3*x2*x1*x3^-1    --->    x0^-1*x1*x0^2\n",
       "    x3*x2*x1*x4^-1    --->    x0^-1*x1*x0*x1\n",
       "    x3*x2^2*x0^-1    --->    x0^-1*x1^2*x2\n",
       "    x3*x2^2*x3^-1    --->    x0^-1*x1^2*x0\n",
       "    x3*x2^2*x4^-1    --->    x0^-1*x1^3\n",
       "    x4^-1*x0^-2*x2^-1    --->    x0^-1*x2^-1*x3^-1*x4^-1\n",
       "    x4*x0^2*x1^-1    --->    x1^-1*x4^2*x3\n",
       "    x4*x0^2*x2^-1    --->    x1^-1*x4^3\n",
       "    x4*x0^2*x3^-1    --->    x1^-1*x4^2*x0\n",
       "    x4*x0^2*x3    --->    x0*x3*x0*x1\n",
       "    x4*x0^2*x4^-1    --->    x1^-1*x4^2*x1\n",
       "    x4*x0*x1*x0^-1    --->    x1^-1*x4*x0*x2\n",
       "    x4*x0*x1*x2^-1    --->    x1^-1*x4*x0*x4\n",
       "    x4*x0*x1*x3^-1    --->    x1^-1*x4*x0^2\n",
       "    x4*x0*x1*x4^-1    --->    x1^-1*x4*x0*x1\n",
       "    x4*x0*x2*x0^-1    --->    x1^-1*x4*x1*x2\n",
       "    x4*x0*x2*x3^-1    --->    x1^-1*x4*x1*x0\n",
       "    x4*x0*x2*x4^-1    --->    x1^-1*x4*x1^2\n",
       "    x4*x1*x0*x1^-1    --->    x1^-1*x0*x4*x3\n",
       "    x4*x1*x0*x2^-1    --->    x1^-1*x0*x4^2\n",
       "    x4*x1*x0*x3^-1    --->    x1^-1*x0*x4*x0\n",
       "    x4*x1*x0*x4^-1    --->    x1^-1*x0*x4*x1\n",
       "    x4*x1^2*x0^-1    --->    x1^-1*x0^2*x2\n",
       "    x4*x1^2*x2^-1    --->    x1^-1*x0^2*x4\n",
       "    x4*x1^2*x3^-1    --->    x1^-1*x0^3\n",
       "    x4*x1^2*x4^-1    --->    x1^-1*x0^2*x1\n",
       "    x4*x1*x2*x0^-1    --->    x1^-1*x0*x1*x2\n",
       "    x4*x1*x2*x1^-1    --->    x1^-1*x0*x1*x3\n",
       "    x4*x1*x2*x3^-1    --->    x1^-1*x0*x1*x0\n",
       "    x4*x1*x2*x4^-1    --->    x1^-1*x0*x1^2\n",
       "    x0^-1*x2^-1*x0^-2*x1    --->    x1^-2*x4^-1\n",
       "    x0^-1*x2^-1*x0^-2*x2    --->    x1^-2*x0^-1\n",
       "    x0^-1*x2^-1*x0^-2*x3    --->    x1^-3\n",
       "    x0^-1*x2^-1*x0^-2*x4    --->    x1^-2*x2^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-1*x4    --->    x1^-1*x2^-2\n",
       "    x0*x3*x0*x2*x1^-1    --->    x4*x1*x3\n",
       "    x0*x3*x2^2*x1^-1    --->    x1^2*x3\n",
       "    x1^-1*x0^-3*x2^-1    --->    x0^-1*x2^-1*x0^-1*x4^-2\n",
       "    x1^-1*x0^3*x3    --->    x4*x1^2\n",
       "    x1*x0^-3*x2^-1    --->    x3^-1*x4^-2\n",
       "    x1*x0^3*x3    --->    x0*x3*x2*x1^2\n",
       "    x2^-1*x0^-3*x2^-1    --->    x0^-1*x2^-1*x1^-1*x4^-2\n",
       "    x2^-1*x0^-1*x1^-1*x0^-1*x2    --->    x0*x1^-1*x2^-1*x1^-1*x0^-1\n",
       "    x2^-1*x0^-1*x1^-1*x0^-1*x3    --->    x0*x1^-1*x2^-1*x1^-2\n",
       "    x2^-1*x0^-1*x1^-1*x0^-1*x4    --->    x0*(x1^-1*x2^-1)^2\n",
       "    x2^-1*x1^-1*x0^-2*x1    --->    x0*x2^-1*x1^-2*x4^-1\n",
       "    x2^-1*x1^-1*x0^-2*x2    --->    x0*x2^-1*x1^-2*x0^-1\n",
       "    x2^-1*x1^-1*x0^-2*x3    --->    x0*x2^-1*x1^-3\n",
       "    x2^-1*x1^-1*x0^-2*x4    --->    x0*x2^-1*x1^-2*x2^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-1*x0    --->    x0*x2^-1*x1^-1*x2^-1*x3^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-1*x2    --->    x0*x2^-1*x1^-1*x2^-1*x0^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-1*x3    --->    x0*(x2^-1*x1^-1)^2\n",
       "    x2^-1*x1^-1*x0^-1*x1^-1*x4    --->    x0*x2^-1*x1^-1*x2^-2\n",
       "    x2^-1*x1^-2*x0^-1*x1    --->    x0*x2^-2*x1^-1*x4^-1\n",
       "    x2^-1*x1^-2*x0^-1*x2    --->    x0*x2^-2*x1^-1*x0^-1\n",
       "    x2^-1*x1^-2*x0^-1*x3    --->    x0*x2^-2*x1^-2\n",
       "    x2^-1*x1^-2*x0^-1*x4    --->    x0*x2^-2*x1^-1*x2^-1\n",
       "    x2*x0^-3*x2^-1    --->    x4^-3\n",
       "    x3^-1*x1^-1*x0^-2*x1    --->    x1*x2^-1*x1^-2*x4^-1\n",
       "    x3^-1*x1^-1*x0^-2*x2    --->    x1*x2^-1*x1^-2*x0^-1\n",
       "    x3^-1*x1^-1*x0^-2*x3    --->    x1*x2^-1*x1^-3\n",
       "    x3^-1*x1^-1*x0^-2*x4    --->    x1*x2^-1*x1^-2*x2^-1\n",
       "    x3^-1*x1^-1*x0^-1*x1^-1*x0    --->    x1*x2^-1*x1^-1*x2^-1*x3^-1\n",
       "    x3^-1*x1^-1*x0^-1*x1^-1*x2    --->    x1*x2^-1*x1^-1*x2^-1*x0^-1\n",
       "    x3^-1*x1^-1*x0^-1*x1^-1*x3    --->    x1*(x2^-1*x1^-1)^2\n",
       "    x3^-1*x1^-1*x0^-1*x1^-1*x4    --->    x1*x2^-1*x1^-1*x2^-2\n",
       "    x3*x0^3*x1^-1    --->    x0^-1*x4^3*x3\n",
       "    x3*x0^3*x2^-1    --->    x0^-1*x4^4\n",
       "    x3*x0^3*x3^-1    --->    x0^-1*x4^3*x0\n",
       "    x3*x0^3*x3    --->    x0*x3*x4*x1^2\n",
       "    x3*x0^3*x4^-1    --->    x0^-1*x4^3*x1\n",
       "    x3*x0^2*x1*x0^-1    --->    x0^-1*x4^2*x0*x2\n",
       "    x3*x0^2*x1*x2^-1    --->    x0^-1*x4^2*x0*x4\n",
       "    x3*x0^2*x1*x3^-1    --->    x0^-1*x4^2*x0^2\n",
       "    x3*x0^2*x1*x4^-1    --->    x0^-1*x4^2*x0*x1\n",
       "    x3*x0^2*x2*x0^-1    --->    x0^-1*x4^2*x1*x2\n",
       "    x3*x0^2*x2*x3^-1    --->    x0^-1*x4^2*x1*x0\n",
       "    x3*x0^2*x2*x4^-1    --->    x0^-1*x4^2*x1^2\n",
       "    x3*x0*x1*x0*x2^-1    --->    x0^-1*x4*x0*x4^2\n",
       "    x3*x0*x1*x0*x4^-1    --->    x0^-1*x4*x0*x4*x1\n",
       "    x3*x0*x2*x1*x0^-1    --->    x0^-1*x4*x1*x0*x2\n",
       "    x3*x0*x2*x1*x2^-1    --->    x0^-1*x4*x1*x0*x4\n",
       "    x3*x0*x2*x1*x3^-1    --->    x0^-1*x4*x1*x0^2\n",
       "    x3*x0*x2*x1*x4^-1    --->    x0^-1*x4*x1*x0*x1\n",
       "    x3*x0*x2^2*x0^-1    --->    x0^-1*x4*x1^2*x2\n",
       "    x3*x0*x2^2*x3^-1    --->    x0^-1*x4*x1^2*x0\n",
       "    x3*x0*x2^2*x4^-1    --->    x0^-1*x4*x1^3\n",
       "    x3*x2*x1*x0*x2^-1    --->    x0^-1*x1*x0*x4^2\n",
       "    x3*x2*x1*x0*x4^-1    --->    x0^-1*x1*x0*x4*x1\n",
       "    x4^-1*x0^-3*x2^-1    --->    x0^-1*x2^-1*x3^-1*x4^-2\n",
       "    x4*x0^3*x3    --->    x0*x3*x0*x1^2\n",
       "    x4*x1*x0^2*x1^-1    --->    x1^-1*x0*x4^2*x3\n",
       "    x4*x1*x0^2*x2^-1    --->    x1^-1*x0*x4^3\n",
       "    x4*x1*x0^2*x3^-1    --->    x1^-1*x0*x4^2*x0\n",
       "    x4*x1*x0^2*x4^-1    --->    x1^-1*x0*x4^2*x1\n",
       "    x4*x1*x0*x1*x0^-1    --->    x1^-1*x0*x4*x0*x2\n",
       "    x4*x1*x0*x1*x2^-1    --->    x1^-1*(x0*x4)^2\n",
       "    x4*x1*x0*x1*x3^-1    --->    x1^-1*x0*x4*x0^2\n",
       "    x4*x1*x0*x1*x4^-1    --->    x1^-1*x0*x4*x0*x1\n",
       "    x4*x1*x0*x2*x0^-1    --->    x1^-1*x0*x4*x1*x2\n",
       "    x4*x1*x0*x2*x3^-1    --->    x1^-1*x0*x4*x1*x0\n",
       "    x4*x1*x0*x2*x4^-1    --->    x1^-1*x0*x4*x1^2\n",
       "    x4*x1^2*x0*x1^-1    --->    x1^-1*x0^2*x4*x3\n",
       "    x4*x1^2*x0*x2^-1    --->    x1^-1*x0^2*x4^2\n",
       "    x4*x1^2*x0*x3^-1    --->    x1^-1*x0^2*x4*x0\n",
       "    x4*x1^2*x0*x4^-1    --->    x1^-1*x0^2*x4*x1\n",
       "    x4*x1^3*x0^-1    --->    x1^-1*x0^3*x2\n",
       "    x4*x1^3*x2^-1    --->    x1^-1*x0^3*x4\n",
       "    x4*x1^3*x3^-1    --->    x1^-1*x0^4\n",
       "    x4*x1^3*x4^-1    --->    x1^-1*x0^3*x1\n",
       "    x4*x1^2*x2*x0^-1    --->    x1^-1*x0^2*x1*x2\n",
       "    x4*x1^2*x2*x1^-1    --->    x1^-1*x0^2*x1*x3\n",
       "    x4*x1^2*x2*x3^-1    --->    x1^-1*x0^2*x1*x0\n",
       "    x4*x1^2*x2*x4^-1    --->    x1^-1*x0^2*x1^2\n",
       "    x0^-1*x2^-1*x0^-1*x1^-1*x0^-1*x1    --->    x1^-1*x2^-1*x1^-1*x4^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-2*x0    --->    x1^-1*x2^-2*x3^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-2*x2    --->    x1^-1*x2^-2*x0^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-2*x3    --->    x1^-1*x2^-2*x1^-1\n",
       "    x0^-1*x2^-1*x1^-3*x0    --->    x2^-3*x3^-1\n",
       "    x0^-1*x2^-1*x1^-3*x2    --->    x2^-3*x0^-1\n",
       "    x0^-1*x2^-1*x1^-3*x3    --->    x2^-3*x1^-1\n",
       "    x0^-1*x2^-2*x0^-2*x1    --->    x3^-1*x1^-2*x4^-1\n",
       "    x0^-1*x2^-2*x0^-2*x2    --->    x3^-1*x1^-2*x0^-1\n",
       "    x0^-1*x2^-2*x0^-2*x3    --->    x3^-1*x1^-3\n",
       "    x0^-1*x2^-2*x0^-2*x4    --->    x3^-1*x1^-2*x2^-1\n",
       "    x0^-1*x2^-2*x0^-1*x1^-1*x4    --->    x3^-1*x1^-1*x2^-2\n",
       "    x0*x3*x0*x1*x0*x1^-1    --->    x4*x0*x4*x3\n",
       "    x0*x3*x0*x1*x0*x3^-1    --->    (x4*x0)^2\n",
       "    x0*x3*x0*x1^2*x0^-1    --->    x4*x0^2*x2\n",
       "    x0*x3*x0*x1^2*x2^-1    --->    x4*x0^2*x4\n",
       "    x0*x3*x0*x1^2*x3^-1    --->    x4*x0^3\n",
       "    x0*x3*x0*x1^2*x4^-1    --->    x4*x0^2*x1\n",
       "    x0*x3*x0*x1*x2*x0^-1    --->    x4*x0*x1*x2\n",
       "    x0*x3*x0*x1*x2*x1^-1    --->    x4*x0*x1*x3\n",
       "    x0*x3*x0*x1*x2*x3^-1    --->    x4*x0*x1*x0\n",
       "    x0*x3*x0*x1*x2*x4^-1    --->    x4*x0*x1^2\n",
       "    x0*x3*x2*x1*x0*x1^-1    --->    x1*x0*x4*x3\n",
       "    x0*x3*x2*x1*x0*x3^-1    --->    x1*x0*x4*x0\n",
       "    x0*x3*x2*x1^2*x0^-1    --->    x1*x0^2*x2\n",
       "    x0*x3*x2*x1^2*x2^-1    --->    x1*x0^2*x4\n",
       "    x0*x3*x2*x1^2*x3^-1    --->    x1*x0^3\n",
       "    x0*x3*x2*x1^2*x4^-1    --->    x1*x0^2*x1\n",
       "    x0*x3*x2*x1*x2*x0^-1    --->    x1*x0*x1*x2\n",
       "    x0*x3*x2*x1*x2*x1^-1    --->    x1*x0*x1*x3\n",
       "    x0*x3*x2*x1*x2*x3^-1    --->    (x1*x0)^2\n",
       "    x0*x3*x2*x1*x2*x4^-1    --->    x1*x0*x1^2\n",
       "    x1^-1*x0^4*x3    --->    x4*x1^3\n",
       "    x1*x0^-4*x2^-1    --->    x3^-1*x4^-3\n",
       "    x1*x0^4*x3    --->    x0*x3*x2*x1^3\n",
       "    x2^-1*x0^-4*x2^-1    --->    x0^-1*x2^-1*x1^-1*x4^-3\n",
       "    x2^-1*x1^-1*x0^-3*x1    --->    x0*x2^-1*x1^-3*x4^-1\n",
       "    x2^-1*x1^-1*x0^-3*x2    --->    x0*x2^-1*x1^-3*x0^-1\n",
       "    x2^-1*x1^-1*x0^-3*x3    --->    x0*x2^-1*x1^-4\n",
       "    x2^-1*x1^-1*x0^-3*x4    --->    x0*x2^-1*x1^-3*x2^-1\n",
       "    x2^-1*x1^-1*x0^-2*x1^-1*x0    --->    x0*x2^-1*x1^-2*x2^-1*x3^-1\n",
       "    x2^-1*x1^-1*x0^-2*x1^-1*x2    --->    x0*x2^-1*x1^-2*x2^-1*x0^-1\n",
       "    x2^-1*x1^-1*x0^-2*x1^-1*x3    --->    x0*x2^-1*x1^-2*x2^-1*x1^-1\n",
       "    x2^-1*x1^-1*x0^-2*x1^-1*x4    --->    x0*x2^-1*x1^-2*x2^-2\n",
       "    x2^-1*(x1^-1*x0^-1)^2*x1    --->    x0*(x2^-1*x1^-1)^2*x4^-1\n",
       "    x2^-1*(x1^-1*x0^-1)^2*x2    --->    x0*(x2^-1*x1^-1)^2*x0^-1\n",
       "    x2^-1*(x1^-1*x0^-1)^2*x3    --->    x0*(x2^-1*x1^-1)^2*x1^-1\n",
       "    x2^-1*(x1^-1*x0^-1)^2*x4    --->    x0*(x2^-1*x1^-1)^2*x2^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-2*x0    --->    x0*x2^-1*x1^-1*x2^-2*x3^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-2*x2    --->    x0*x2^-1*x1^-1*x2^-2*x0^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-2*x3    --->    x0*x2^-1*x1^-1*x2^-2*x1^-1\n",
       "    x2^-1*x1^-1*x0^-1*x1^-2*x4    --->    x0*x2^-1*x1^-1*x2^-3\n",
       "    x2*x0^-4*x2^-1    --->    x4^-4\n",
       "    x3*x0^4*x3    --->    x0*x3*x4*x1^3\n",
       "    x4^-1*x0^-4*x2^-1    --->    x0^-1*x2^-1*x3^-1*x4^-3\n",
       "    x4*x0^4*x3    --->    x0*x3*x0*x1^3\n",
       "    x4*x1^2*x0^2*x1^-1    --->    x1^-1*x0^2*x4^2*x3\n",
       "    x4*x1^2*x0^2*x2^-1    --->    x1^-1*x0^2*x4^3\n",
       "    x4*x1^2*x0^2*x3^-1    --->    x1^-1*x0^2*x4^2*x0\n",
       "    x4*x1^2*x0^2*x4^-1    --->    x1^-1*x0^2*x4^2*x1\n",
       "    x4*x1^2*x0*x1*x0^-1    --->    x1^-1*x0^2*x4*x0*x2\n",
       "    x4*x1^2*x0*x1*x2^-1    --->    x1^-1*x0*(x0*x4)^2\n",
       "    x4*x1^2*x0*x1*x3^-1    --->    x1^-1*x0^2*x4*x0^2\n",
       "    x4*x1^2*x0*x1*x4^-1    --->    x1^-1*x0^2*x4*x0*x1\n",
       "    x4*x1^2*x0*x2*x0^-1    --->    x1^-1*x0^2*x4*x1*x2\n",
       "    x4*x1^2*x0*x2*x3^-1    --->    x1^-1*x0^2*x4*x1*x0\n",
       "    x4*x1^2*x0*x2*x4^-1    --->    x1^-1*x0^2*x4*x1^2\n",
       "    x4*x1^3*x0*x1^-1    --->    x1^-1*x0^3*x4*x3\n",
       "    x4*x1^3*x0*x2^-1    --->    x1^-1*x0^3*x4^2\n",
       "    x4*x1^3*x0*x3^-1    --->    x1^-1*x0^3*x4*x0\n",
       "    x4*x1^3*x0*x4^-1    --->    x1^-1*x0^3*x4*x1\n",
       "    x4*x1^4*x0^-1    --->    x1^-1*x0^4*x2\n",
       "    x4*x1^4*x2^-1    --->    x1^-1*x0^4*x4\n",
       "    x4*x1^4*x3^-1    --->    x1^-1*x0^5\n",
       "    x4*x1^4*x4^-1    --->    x1^-1*x0^4*x1\n",
       "    x4*x1^3*x2*x0^-1    --->    x1^-1*x0^3*x1*x2\n",
       "    x4*x1^3*x2*x1^-1    --->    x1^-1*x0^3*x1*x3\n",
       "    x4*x1^3*x2*x3^-1    --->    x1^-1*x0^3*x1*x0\n",
       "    x4*x1^3*x2*x4^-1    --->    x1^-1*x0^3*x1^2\n",
       "    x0^-2*x2^-1*x1^-3*x4    --->    x0^-1*x2^-4\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-1*x1    --->    x0^-1*x2^-1*x0^-1*x1^-1*x4^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-1*x0^-2*x1    --->    x1^-1*x2^-1*x1^-2*x4^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-1*x0^-2*x2    --->    x1^-1*x2^-1*x1^-2*x0^-1\n",
       "    x0^-1*x2^-1*x0^-1*x1^-1*x0^-2*x3    --->    x1^-1*x2^-1*x1^-3\n",
       "    x0^-1*x2^-1*x0^-1*x1^-1*x0^-2*x4    --->    x1^-1*x2^-1*x1^-2*x2^-1\n",
       "    x0^-1*x2^-1*(x0^-1*x1^-1)^2*x0    --->    (x1^-1*x2^-1)^2*x3^-1\n",
       "    x0^-1*x2^-1*(x0^-1*x1^-1)^2*x2    --->    (x1^-1*x2^-1)^2*x0^-1\n",
       "    x0^-1*x2^-1*(x0^-1*x1^-1)^2*x3    --->    (x1^-1*x2^-1)^2*x1^-1\n",
       "    x0^-1*x2^-1*(x0^-1*x1^-1)^2*x4    --->    (x1^-1*x2^-1)^2*x2^-1\n",
       "    x0^-1*x2^-1*x1^-2*x0^-2*x1    --->    x2^-2*x1^-2*x4^-1\n",
       "    x0^-1*x2^-1*x1^-2*x0^-2*x2    --->    x2^-2*x1^-2*x0^-1\n",
       "    x0^-1*x2^-1*x1^-2*x0^-2*x3    --->    x2^-2*x1^-3\n",
       "    x0^-1*x2^-1*x1^-2*x0^-2*x4    --->    x2^-2*x1^-2*x2^-1\n",
       "    x0^-1*x2^-1*x1^-2*x0^-1*x1^-1*x0    --->    x2^-2*x1^-1*x2^-1*x3^-1\n",
       "    x0^-1*x2^-1*x1^-2*x0^-1*x1^-1*x2    --->    x2^-2*x1^-1*x2^-1*x0^-1\n",
       "    x0^-1*x2^-1*x1^-2*x0^-1*x1^-1*x3    --->    x2^-1*(x2^-1*x1^-1)^2\n",
       "    x0^-1*x2^-1*x1^-2*x0^-1*x1^-1*x4    --->    x2^-2*x1^-1*x2^-2\n",
       "    x0*x1^-1*x0^-4*x2^-1    --->    x2^-1*x0^-1*x4^-3\n",
       "    x0*x3*x0*x1*x0^2*x2^-1    --->    x4*x0*x4^3\n",
       "    x0*x3*x0*x1*x0^2*x3^-1    --->    x4*x0*x4^2*x0\n",
       "    x0*x3*x0*x1*x0^2*x4^-1    --->    x4*x0*x4^2*x1\n",
       "    x0*x3*x0*x1*x0*x2*x0^-1    --->    x4*x0*x4*x1*x2\n",
       "    x0*x3*x0*x1*x0*x2*x3^-1    --->    x4*x0*x4*x1*x0\n",
       "    x0*x3*x0*x1*x0*x2*x4^-1    --->    x4*x0*x4*x1^2\n",
       "    x0*x3*x0*x1^2*x0*x1^-1    --->    x4*x0^2*x4*x3\n",
       "    x0*x3*x0*x1^2*x0*x2^-1    --->    x4*x0^2*x4^2\n",
       "    x0*x3*x0*x1^2*x0*x3^-1    --->    x4*x0^2*x4*x0\n",
       "    x0*x3*x0*x1^2*x0*x4^-1    --->    x4*x0^2*x4*x1\n",
       "    x0*x3*x0*x1^2*x2*x0^-1    --->    x4*x0^2*x1*x2\n",
       "    x0*x3*x0*x1^2*x2*x1^-1    --->    x4*x0^2*x1*x3\n",
       "    x0*x3*x0*x1^2*x2*x3^-1    --->    x4*x0^2*x1*x0\n",
       "    x0*x3*x0*x1^2*x2*x4^-1    --->    x4*x0^2*x1^2\n",
       "    x0*x3*x2*x1*x0^2*x2^-1    --->    x1*x0*x4^3\n",
       "    x0*x3*x2*x1*x0^2*x3^-1    --->    x1*x0*x4^2*x0\n",
       "    x0*x3*x2*x1*x0^2*x4^-1    --->    x1*x0*x4^2*x1\n",
       "    x0*x3*x2*x1*x0*x2*x0^-1    --->    x1*x0*x4*x1*x2\n",
       "    x0*x3*x2*x1*x0*x2*x3^-1    --->    x1*x0*x4*x1*x0\n",
       "    x0*x3*x2*x1*x0*x2*x4^-1    --->    x1*x0*x4*x1^2\n",
       "    x0*x3*x2*x1^2*x0*x1^-1    --->    x1*x0^2*x4*x3\n",
       "    x0*x3*x2*x1^2*x0*x2^-1    --->    x1*x0^2*x4^2\n",
       "    x0*x3*x2*x1^2*x0*x3^-1    --->    x1*x0^2*x4*x0\n",
       "    x0*x3*x2*x1^2*x0*x4^-1    --->    x1*x0^2*x4*x1\n",
       "    x0*x3*x2*x1^2*x2*x0^-1    --->    x1*x0^2*x1*x2\n",
       "    x0*x3*x2*x1^2*x2*x1^-1    --->    x1*x0^2*x1*x3\n",
       "    x0*x3*x2*x1^2*x2*x3^-1    --->    x1*x0^2*x1*x0\n",
       "    x0*x3*x2*x1^2*x2*x4^-1    --->    x1*x0^2*x1^2\n",
       "    x1^-1*x0^5*x3    --->    x4*x1^4\n",
       "    x1*x0^-5*x2^-1    --->    x3^-1*x4^-4\n",
       "    x2^-1*x0^-5*x2^-1    --->    x0^-1*x2^-1*x1^-1*x4^-4\n",
       "    x2^-1*x1^-1*x0^-4*x2^-1    --->    x0^-1*x2^-1*x1^-1*x0^-1*x4^-3\n",
       "    x2*x0^-5*x2^-1    --->    x4^-5\n",
       "    x4*x1^3*x0^2*x1^-1    --->    x1^-1*x0^3*x4^2*x3\n",
       "    x4*x1^3*x0^2*x2^-1    --->    x1^-1*x0^3*x4^3\n",
       "    x4*x1^3*x0^2*x3^-1    --->    x1^-1*x0^3*x4^2*x0\n",
       "    x4*x1^3*x0^2*x4^-1    --->    x1^-1*x0^3*x4^2*x1\n",
       "    x4*x1^3*x0*x2*x0^-1    --->    x1^-1*x0^3*x4*x1*x2\n",
       "    x4*x1^3*x0*x2*x3^-1    --->    x1^-1*x0^3*x4*x1*x0\n",
       "    x4*x1^3*x0*x2*x4^-1    --->    x1^-1*x0^3*x4*x1^2\n",
       "    x4*x1^4*x0*x1^-1    --->    x1^-1*x0^4*x4*x3\n",
       "    x4*x1^4*x0*x2^-1    --->    x1^-1*x0^4*x4^2\n",
       "    x4*x1^4*x0*x3^-1    --->    x1^-1*x0^4*x4*x0\n",
       "    x4*x1^4*x0*x4^-1    --->    x1^-1*x0^4*x4*x1\n",
       "    x4*x1^4*x2*x0^-1    --->    x1^-1*x0^4*x1*x2\n",
       "    x4*x1^4*x2*x1^-1    --->    x1^-1*x0^4*x1*x3\n",
       "    x4*x1^4*x2*x3^-1    --->    x1^-1*x0^4*x1*x0\n",
       "    x4*x1^4*x2*x4^-1    --->    x1^-1*x0^4*x1^2\n",
       "    x0^-2*x2^-1*x1^-1*x2^-1*x0^-2*x1    --->    x0^-1*x2^-1*x3^-1*x1^-2*x4^-1\n",
       "    x0^-2*x2^-1*x1^-1*x2^-1*x0^-2*x2    --->    x0^-1*x2^-1*x3^-1*x1^-2*x0^-1\n",
       "    x0^-2*x2^-1*x1^-1*x2^-1*x0^-2*x3    --->    x0^-1*x2^-1*x3^-1*x1^-3\n",
       "    x0^-2*x2^-1*x1^-1*x2^-1*x0^-2*x4    --->    x0^-1*x2^-1*x3^-1*x1^-2*x2^-1\n",
       "    x0^-2*x2^-1*x1^-1*x2^-1*x0^-1*x1^-1*x4    --->    x0^-1*x2^-1*x3^-1*x1^-1*x2^-2\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-2*x1    --->    x0^-1*x2^-1*x0^-1*x1^-2*x4^-1\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-2*x2    --->    x0^-1*x2^-1*x0^-1*x1^-2*x0^-1\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-2*x3    --->    x0^-1*x2^-1*x0^-1*x1^-3\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-2*x4    --->    x0^-1*x2^-1*x0^-1*x1^-2*x2^-1\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-1*x1^-1*x0    --->    x0^-1*x2^-1*x0^-1*x1^-1*x2^-1*x3^-1\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-1*x1^-1*x2    --->    x0^-1*x2^-1*x0^-1*x1^-1*x2^-1*x0^-1\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-1*x1^-1*x3    --->    x0^-1*x2^-1*x0^-1*x1^-1*x2^-1*x1^-1\n",
       "    x0^-2*x2^-1*x1^-1*x4^-1*x0^-1*x1^-1*x4    --->    x0^-1*x2^-1*x0^-1*x1^-1*x2^-2\n",
       "    x0^-1*x3*x0^5*x3    --->    x3*x4*x1^4\n",
       "    x0*x1^-1*x0^-5*x2^-1    --->    x2^-1*x0^-1*x4^-4\n",
       "    x0*x3*x0*x1^2*x0^2*x2^-1    --->    x4*x0^2*x4^3\n",
       "    x0*x3*x0*x1^2*x0^2*x3^-1    --->    x4*x0^2*x4^2*x0\n",
       "    x0*x3*x0*x1^2*x0^2*x4^-1    --->    x4*x0^2*x4^2*x1\n",
       "    x0*x3*x0*x1^2*x0*x2*x0^-1    --->    x4*x0^2*x4*x1*x2\n",
       "    x0*x3*x0*x1^2*x0*x2*x3^-1    --->    x4*x0^2*x4*x1*x0\n",
       "    x0*x3*x0*x1^2*x0*x2*x4^-1    --->    x4*x0^2*x4*x1^2\n",
       "    x0*x3*x2*x1^2*x0^2*x2^-1    --->    x1*x0^2*x4^3\n",
       "    x0*x3*x2*x1^2*x0^2*x3^-1    --->    x1*x0^2*x4^2*x0\n",
       "    x0*x3*x2*x1^2*x0^2*x4^-1    --->    x1*x0^2*x4^2*x1\n",
       "    x0*x3*x2*x1^2*x0*x2*x0^-1    --->    x1*x0^2*x4*x1*x2\n",
       "    x0*x3*x2*x1^2*x0*x2*x3^-1    --->    x1*x0^2*x4*x1*x0\n",
       "    x0*x3*x2*x1^2*x0*x2*x4^-1    --->    x1*x0^2*x4*x1^2\n",
       "    x0*x4^-1*x0^-5*x2^-1    --->    x2^-1*x3^-1*x4^-4\n",
       "    x1^-1*x0^6*x3    --->    x4*x1^5\n",
       "    x1^-1*x3*x0^5*x3    --->    x4^2*x1^4\n",
       "    x1*x0^-6*x2^-1    --->    x3^-1*x4^-5\n",
       "    x1*x4^-1*x0^-5*x2^-1    --->    x3^-2*x4^-4\n",
       "    x2^-1*x1^-1*x0^-5*x2^-1    --->    x0^-1*x2^-1*x1^-1*x0^-1*x4^-4\n",
       "    x2*x0^-6*x2^-1    --->    x4^-6\n",
       "    x2*x4^-1*x0^-5*x2^-1    --->    x4^-1*x3^-1*x4^-4\n",
       "    x4*x1^4*x0^2*x2^-1    --->    x1^-1*x0^4*x4^3\n",
       "    x4*x1^4*x0^2*x3^-1    --->    x1^-1*x0^4*x4^2*x0\n",
       "    x4*x1^4*x0^2*x4^-1    --->    x1^-1*x0^4*x4^2*x1\n",
       "    x4*x1^4*x0*x2*x0^-1    --->    x1^-1*x0^4*x4*x1*x2\n",
       "    x4*x1^4*x0*x2*x3^-1    --->    x1^-1*x0^4*x4*x1*x0\n",
       "    x4*x1^4*x0*x2*x4^-1    --->    x1^-1*x0^4*x4*x1^2\n",
       "    x0^-3*x2^-1*x1^-1*x0^-1*x4^-1*x0^-1*x1    --->    x0^-2*x2^-1*x1^-1*x0^-1*x1^-1*x4^-1\n",
       "    x0^-1*x1*x3*x0^5*x3    --->    x3*x2*x4*x1^4\n",
       "    x0^-1*x4*x3*x0^5*x3    --->    x3*x0*x4*x1^4\n",
       "    x0*x1^-1*x0^-6*x2^-1    --->    x2^-1*x0^-1*x4^-5\n",
       "    x0*x2^-1*x0^-6*x2^-1    --->    x2^-1*x1^-1*x4^-5\n",
       "    x0*x2^-1*x1^-2*x0^-4*x2^-1    --->    x2^-1*x1^-1*x0^-2*x4^-3\n",
       "    x1*x2^-1*x0^-6*x2^-1    --->    x3^-1*x1^-1*x4^-5\n",
       "    x0*x2^-1*x1^-1*x0^-6*x2^-1    --->    x2^-1*x1^-1*x0^-1*x4^-5\n",
       "    x0*x2^-1*x1^-2*x0^-5*x2^-1    --->    x2^-1*x1^-1*x0^-2*x4^-4\n",
       "    x0*x2^-1*x1^-1*x4^-1*x0^-5*x2^-1    --->    x2^-1*x1^-1*x0^-1*x3^-1*x4^-4\n",
       "    x0*x2^-2*x0^-6*x2^-1    --->    x2^-1*x1^-2*x4^-5\n",
       "    x1*x2^-1*x1^-1*x0^-6*x2^-1    --->    x3^-1*x1^-1*x0^-1*x4^-5\n",
       "    x1*x2^-1*x1^-1*x4^-1*x0^-5*x2^-1    --->    x3^-1*x1^-1*x0^-1*x3^-1*x4^-4\n",
       "    x0*x2^-1*x1^-2*x0^-6*x2^-1    --->    x2^-1*x1^-1*x0^-2*x4^-5\n",
       "    x0*x2^-1*x1^-2*x4^-1*x0^-5*x2^-1    --->    x2^-1*x1^-1*x0^-2*x3^-1*x4^-4\n",
       "    x0*x2^-1*x1^-1*x2^-1*x0^-6*x2^-1    --->    x2^-1*x1^-1*x0^-1*x1^-1*x4^-5"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Defining x0, x1, x2\n"
     ]
    }
   ],
   "source": [
    "G.inject_variables()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x2*x1*x0"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.reduce(x2*x1*x0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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