Function Repository Resource:

PoincarePolynomial

Source Notebook

Compute the Poincaré polynomial for a simplicial complex

Contributed by: Richard Hennigan (Wolfram Research)

ResourceFunction["PoincarePolynomial"][cplx]

gives the Poincaré polynomial for the specified simplicial complex cplx.

Details and Options

In ResourceFunction["PoincarePolynomial"][cplx], the value for cplx can be any of the following:

simplex

a simplex as defined in the table below

{simplex₁,simplex₂,…}

a list of simplices

{{v₁₁,v₁₂,…},…}

a list of lists, where the v_i,j correspond to simplex vertices

Graph[…]

a graph object

MeshRegion[…]

a mesh region

The value for simplex can be any of the following:

Point[v]

a point

Line[{v₁,v₂}]

a line segment

Triangle[{v₁,v₂,v₃}] or Polygon[{v₁,v₂,v₃}]

a filled triangle

Tetrahedron[{v₁,v₂,v₃,v₄}]

a filled tetrahedron

Simplex[{v₁,…,v_n}]

an n-1 dimensional simplex

Examples

Basic Examples (6)

Get the Poincaré polynomial for a simplex:

In[1]:=

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Get the Poincaré polynomial for a simplicial complex:

In[2]:=

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Get the Poincaré polynomial for a MeshRegion:

In[3]:=

$ResourceFunction["PoincarePolynomial"][\!\(\* GraphicsBox[ TagBox[ DynamicModuleBox[{Typeset`mesh = {MeshRegion, {}}}, TagBox[GraphicsComplexBox[CompressedData[" 1:eJxdVM9rU0EQfjaRqgmIIGJTECy0UVBE8CTY2b9A1FPBU4OC6EHQNiqR1h8n D0oh2PYSBC+ioFQQMbQyK6gHI15qy7MIBpFarYWkohRp1Pfcbx7uhBc2k8n3 Y77d7PbC6SPH24Ig6Ive8Rq9tu7uNL0rheEzh9p/8NnqnU2jpXt8MdcfffrJ mXLu9s6HIS8NrF7NlH/xjmIhXyx85kp4LXpavLp5LPts/zLv6bgy+OjCH3aU K1gDm1o88b33RSvpn4/h+TYr+Fv3p2pzX9J2Efydsexouy1Bf3Km4+740fX2 bbxUmnzw2/Po2WA/xPDKAo8PnNz26UHGXh6ZHHrc9w66WQv9QPqXXJ8EX3d4 qoJ/2vGT0iflj875/knNR2p+0/LzoW4/P1qLfIWfkL/oRxxd/+2P9En6wJPg uxw/CT/0Sfkj8Qf/SR/7YwSP+Y3wIx8j+sjPzCI/5GtmkC/yN9gf6GaN7M8Y +iXsn8Kz4ucc9OV8ij85n4O+f07787Ga36p8uNvPj1W+rPIHX134gmOxfPEj ufMZ8qsD1yfeTy/g9/Psvm6Cf0n5acD/76R/I0Kvq63BvPPc889fCvmEwKeT PPN+nwQPfexvwq/OQ4OUP6r5/knmq4Mf88PHyFS/67PoY34WPPhZ6bPoY/6k D/9W8O7+SVlfP5383zG/lfwVHvdPwq/uqwYrf/wa/iV/NR/vO5yZ4/IbftK8 OXFqdpk3oq66mvYO9XyNasqFL5/G9S7UW1zNfwH58ZfD "], {Hue[0.6, 0.3, 0.95], EdgeForm[Hue[0.6, 0.3, 0.75]], TagBox[PolygonBox[{{79, 11, 10}, {80, 9, 8}, {79, 10, 9}, {7, 80, 8}, {13, 12, 78}, {78, 12, 11}, {78, 11, 79}, {9, 80, 79}, {77, 13, 78}, {7, 81, 80}, {81, 6, 5}, {82, 4, 3}, { 28, 27, 87}, {82, 5, 4}, {85, 3, 2}, {81, 5, 82}, {3, 85, 82}, {69, 84, 53}, {84, 83, 85}, {53, 70, 69}, {1, 88, 85}, {2, 1, 85}, {6, 81, 7}, {15, 14, 77}, {15, 76, 16}, { 77, 76, 15}, {14, 13, 77}, {17, 76, 75}, {16, 76, 17}, {75, 18, 17}, {75, 19, 18}, {74, 20, 19}, {19, 75, 74}, {74, 73, 21}, {26, 86, 27}, {71, 86, 72}, {73, 72, 23}, {73, 23, 22}, {22, 21, 73}, {24, 23, 72}, {25, 24, 72}, {25, 86, 26}, {25, 72, 86}, {70, 68, 86}, {74, 21, 20}, {84, 85, 54}, {51, 56, 88}, {70, 86, 71}, {56, 50, 49}, {51, 50, 56}, {49, 48, 57}, {53, 84, 54}, {55, 88, 56}, {57, 56, 49}, {54, 85, 88}, {48, 47, 57}, {68, 70, 53}, {58, 47, 46}, {59, 45, 44}, {45, 58, 46}, {43, 59, 44}, {58, 45, 59}, {59, 43, 60}, {60, 41, 61}, {41, 60, 42}, {61, 41, 40}, {42, 60, 43}, {47, 58, 57}, {68, 67, 87}, {68, 87, 86}, {87, 66, 29}, {87, 27, 86}, {31, 66, 65}, {87, 67, 66}, {87, 29, 28}, {66, 30, 29}, {66, 31, 30}, {32, 31, 65}, {54, 88, 55}, {65, 33, 32}, {85, 83, 82}, {37, 63, 62}, {51, 88, 52}, {62, 39, 38}, {62, 61, 39}, {37, 62, 38}, {33, 64, 34}, {64, 33, 65}, {34, 64, 35}, {63, 35, 64}, {63, 36, 35}, {63, 37, 36}, {40, 39, 61}, {52, 88, 1}}], Annotation[#, "Geometry"]& ]}], MouseAppearanceTag["LinkHand"]], AllowKernelInitialization->False], "MeshGraphics", AutoDelete->True, Editable->False, Selectable->False], DefaultBaseStyle->{"MeshGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.1, 1, 0.7]}]\)]$

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Specify the variable:

In[4]:=

$ResourceFunction["PoincarePolynomial"][\!\(\* Graphics3DBox[ TagBox[ DynamicModuleBox[{Typeset`mesh = {MeshRegion, {}}}, TagBox[GraphicsComplex3DBox[CompressedData[" 1:eJyFmG9olWUYxo9OndP+rOPZzmnMLb8kDSxISxrD94EGQRt92KxOQbTKalCY iKBk1hJxEiiM0AZFSfVhoUVkfqgkU4kWZrDICkXo7wjLD9ZsYVMKzvm9L+/v 3Wgwxntz736u+7nv63qe+1ny8FM9j87O5XKj//3W5PipD5W/1wW+V188fuKj 84Uw0PZ63eDK1pD2qw+7L6//c21Ha+jet6vv5/vy4fA/5VPdBwvhm/aD5R1b WmJ72n9x2PtC/9EfTiwKG555ra+hWAjbLwxsfGlPc2zH/4lZO68YOtYU2xds zq/e1rsotN/5R+OBjqYZ/EvCkw9n53x8y9G7Sxk8lb+l+Lvt0i/jZzblQ2gY 23bs5WImX+L3v/XjqpXdCzPxsRv/J2dv3NyxZn4GP3bvD3bvz/T+rcKT7D/2 m95u3zr0+LXVejbH/7/8ju9nvzpRquJqjvdn5/nhr/Z9VqzGS/b5tvzhde/0 Nlbtpdj+4qGN4dD1DVW8xTjO8D2Lj4weKFT3tTFe1/7gdHzyNR7sxk8c8iRv 1sWP/0v3SVO8DnmBA1zpvi3GeRCHPMmbde2frlcSn7yMJ90/CX7iwE/4yrr0 AX0BTvqJ/iIv+pI+xU5/0+/EgSfwhnXtD07HJy/jwW78aX4lOmU+8A1OeIWd vODn9PZSJo71wrrANzgdn7yMJ21vzcShvuWF/Wt2NM/L1Hfr/Lu6Hts9L1Pf Yu2RVSfHajP1PVd3+t7Jcl2mvpU4CzL13T93xc2dVyW6Zn/Xl/iuL3hcX/Cb v51dDZMfPHQxMn9X/PTsxOCyqcj8HW/+8v6RdbkMf3edfr7ljXJNhr+nttc/ N/je3Ax/x3o+3bJ2ONln+5u/xDd/wWP+gt/63Lds79CS9b/G+cLz95c++eCZ R85F1uc517yy54u2C5H1uWK/FFmfK/s/K6PPlfrWZPQZf+sz8a3P4LE+gz9d 30Kof/fD4y1Xfx7nC89HevZveHrpd3G+6MLXvX/f3tn1W+T7QNO3t765/MrJ KF33fCj/XnvD5MjlKF3ffBg4+cBfm0ZzIb1u4g9Ox/e5DJ503RP8vg/kUj+J flTqPhr5flLh+3jke05l/yZmsE9l4lQXzOgW/sZJfN83wGM7+ImDTrm+8Nz1 RRdcX3TE9UV3XF90yvW1Pzgdn7yMB7vxEwedMn/hufmLLpi/6Ij5i+6Yv+iU +Wt/cDo+eRkPduMnDnlan/GzPhPX+gwO6zO4rc/kaX22Pzgdn7yMB7vxE4c+ 9vlLH/j8pW98/tJnPn/pS5+/9LHPX/uD0/HJy3iwG39aB2bWKb7BiS5gJy90 ZHr7VCbO/+kU3+lzJIlPXsaD3fhdX9+f2R/fn9lP35/Zf9+fqZfvz9TX92f7 u76+PxuP6+v7M/z1fET/ez6CL56P4JfnI/jo+Qj+ej6yv/nr+ch4zF/PR+iU 51947vkXXfD8i454/kV3PP+iU55/7W999vxrPNZnz7/olN9t4LnfZ9AFv8Og I35vQXf8roJO+f3E/j5//R5iPD5//b7xL0+XFdQ= "], {Hue[0.6, 0.3, 0.85], EdgeForm[Hue[0.6, 0.3, 0.75]], TagBox[Polygon3DBox[CompressedData[" 1:eJwBwQQ++yFib1JiAgAAAJABAAADAAAAAQIDAwIEAwQFBQQGBQYHBwYIBwgJ CQgKCQoLCwoMCwwNDQwODQ4PDw4QDxARERASERITExIUExQBARQCAhUEBBUW BBYGBhYXBhcICBcYCBgKChgZChkMDBkaDBoODhobDhsQEBscEBwSEhwdEh0U FB0eFB4CAh4VFR8WFh8gFiAXFyAhFyEYGCEiGCIZGSIjGSMaGiMkGiQbGyQl GyUcHCUmHCYdHSYnHSceHicoHigVFSgfHykgICkqICohISorISsiIissIiwj IywtIy0kJC0uJC4lJS4vJS8mJi8wJjAnJzAxJzEoKDEyKDIfHzIpKTMqKjM0 KjQrKzQ1KzUsLDU2LDYtLTY3LTcuLjc4LjgvLzg5LzkwMDk6MDoxMTo7MTsy Mjs8MjwpKTwzMz00ND0+ND41NT4/NT82Nj9ANkA3N0BBN0E4OEFCOEI5OUJD OUM6OkNEOkQ7O0RFO0U8PEVGPEYzM0Y9PUc+PkdIPkg/P0hJP0lAQElKQEpB QUpLQUtCQktMQkxDQ0xNQ01ERE1ORE5FRU5PRU9GRk9QRlA9PVBHR1FISFFS SFJJSVJTSVNKSlNUSlRLS1RVS1VMTFVWTFZNTVZXTVdOTldYTlhPT1hZT1lQ UFlaUFpHR1pRUVtSUltcUlxTU1xdU11UVF1eVF5VVV5fVV9WVl9gVmBXV2Bh V2FYWGFiWGJZWWJjWWNaWmNkWmRRUWRbW2VcXGVmXGZdXWZnXWdeXmdoXmhf X2hpX2lgYGlqYGphYWprYWtiYmtsYmxjY2xtY21kZG1uZG5bW25lZW9mZm9w ZnBnZ3BxZ3FoaHFyaHJpaXJzaXNqanN0anRra3R1a3VsbHV2bHZtbXZ3bXdu bnd4bnhlZXhvb3lwcHl6cHpxcXp7cXtycnt8cnxzc3x9c310dH1+dH51dX5/ dX92dn+AdoB3d4CBd4F4eIGCeIJvb4J5eYN6eoOEeoR7e4SFe4V8fIWGfIZ9 fYaHfYd+foeIfoh/f4iJf4mAgImKgIqBgYqLgYuCgouMgox5eYyDg42EhI2O hI6FhY6PhY+Gho+QhpCHh5CRh5GIiJGSiJKJiZKTiZOKipOUipSLi5SVi5WM jJWWjJaDg5aNjZeOjpeYjpiPj5iZj5mQkJmakJqRkZqbkZuSkpuckpyTk5yd k52UlJ2elJ6VlZ6flZ+Wlp+glqCNjaCXl6GYmKGimKKZmaKjmaOamqOkmqSb m6Slm6WcnKWmnKadnaannaeenqeonqifn6ipn6mgoKmqoKqXl6qhoauioqus oqyjo6yto62kpK2upK6lpa6vpa+mpq+wprCnp7Cxp7GoqLGyqLKpqbKzqbOq qrO0qrShobSrq7WsrLW2rLatrba3rbeurre4rrivr7i5r7mwsLm6sLqxsbq7 sbuysru8sryzs7y9s720tL2+tL6rq761tb+2tr/AtsC3t8DBt8G4uMHCuMK5 ucLDucO6usPEusS7u8TFu8W8vMXGvMa9vcbHvce+vsfIvsi1tci/vwHAwAED wAPBwQMFwQXCwgUHwgfDwwcJwwnExAkLxAvFxQsNxQ3Gxg0Pxg/Hxw8RxxHI yBETyBO/vxMBxinZZA== "]], Annotation[#, "Geometry"]& ]}], MouseAppearanceTag["LinkHand"]], AllowKernelInitialization->False], "MeshGraphics3D", AutoDelete->True, Editable->False, Selectable->False], Boxed->False, DefaultBaseStyle->{"MeshGraphics3D", FrontEnd`GraphicsHighlightColor -> Hue[0.1, 1, 0.7]}, ImageSize->{90., 62.419354838709694`}, Lighting->{{"Ambient", GrayLevel[0.45]}, {"Directional", GrayLevel[0.3], ImageScaled[{2, 0, 2}]}, {"Directional", GrayLevel[0.33], ImageScaled[{2, 2, 2}]}, {"Directional", GrayLevel[0.3], ImageScaled[{0, 2, 2}]}}, Method->{"ShrinkWrap" -> True}, ViewPoint->{1.3, -2.4, 2.}, ViewVertical->{0., 0., 1.}]\), y]$

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Obtain the Poincaré polynomial as a Function:

In[5]:=

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Obtain a Series:

In[6]:=

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Scope (3)

Many graphics primitives that represent simplices can be used:

In[7]:=

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In[8]:=

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In[9]:=

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A simplicial complex can be specified as lists of indices:

In[10]:=

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Get the Poincaré polynomial for a graph:

In[11]:=

$ResourceFunction["PoincarePolynomial"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {Null, {{1, 4}, {1, 5}, {1, 2}, {1, 10}, {2, 5}, {3, 9}, {5, 10}, {6, 7}, {6, 8}, {7, 8}}}, {VertexCoordinates -> {{0.1379861045317623, 0.5054785042100323}, {0.30085516318311756`, 0.24811944512352757`}, {0.9643725633170959, 0.281893794461469}, {0.013410792347853917`, 0.4517414533524742}, {0.3512562295127406, 0.54475760567341}, {0.8149738215243276, 0.9151438616469425}, {0.7798230684079481, 0.7402705338690392}, {0.8489797390389588, 0.6518629743349105}, {0.9346013239549271, 0.12530759475004105`}, {0.3659337392098154, 0.7186686308332282}}}]]}, TagBox[GraphicsGroupBox[ GraphicsComplexBox[{{0.1379861045317623, 0.5054785042100323}, {0.30085516318311756`, 0.24811944512352757`}, {0.9643725633170959, 0.281893794461469}, {0.013410792347853917`, 0.4517414533524742}, {0.3512562295127406, 0.54475760567341}, {0.8149738215243276, 0.9151438616469425}, {0.7798230684079481, 0.7402705338690392}, {0.8489797390389588, 0.6518629743349105}, {0.9346013239549271, 0.12530759475004105`}, {0.3659337392098154, 0.7186686308332282}}, { {Hue[0.6, 0.7, 0.5], Opacity[0.7], LineBox[{{1, 2}, {1, 4}, {1, 5}, {1, 10}, {2, 5}, {3, 9}, { 5, 10}, {6, 7}, {6, 8}, {7, 8}}]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.01210574334443845], DiskBox[2, 0.01210574334443845], DiskBox[3, 0.01210574334443845], DiskBox[4, 0.01210574334443845], DiskBox[5, 0.01210574334443845], DiskBox[6, 0.01210574334443845], DiskBox[7, 0.01210574334443845], DiskBox[8, 0.01210574334443845], DiskBox[9, 0.01210574334443845], DiskBox[10, 0.01210574334443845]}}]], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None]\)]$

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Properties and Relations (1)

The coefficient of x^k in the Poincaré polynomial corresponds to the Betti number b_k:

In[12]:=

(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/82644c70-51f9-47bf-ba7f-49bdf226a40f"]

In[13]:=

$Table[ With[{p = ResourceFunction["PoincarePolynomial"][mesh]}, <| "Mesh" -> mesh, "PoincarePolynomial" -> p, "BettiNumbers" -> ResourceFunction["BettiNumbers"][mesh], "CoefficientList" -> CoefficientList[p, \[FormalX]] |>], {mesh, meshes}] // ResourceFunction["PrettyGrid"]$

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Possible Issues (5)

PoincarePolynomial only considers unique vertices in each simplex:

In[14]:=

In[15]:=

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In[16]:=

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As graphs, these are considered distinct:

In[17]:=

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In[18]:=

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In[19]:=

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The same is true for hypergraphs:

In[20]:=

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All graphs are treated as undirected:

In[21]:=

$ResourceFunction["PoincarePolynomial"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5}, {{{1, 2}, {2, 3}, {3, 1}, {3, 4}, {4, 5}, {3, 5}}, Null}, {EdgeStyle -> { Arrowheads[Small]}, VertexSize -> {Medium}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Arrowheads[0.03374070617041407], Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$1", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], TagBox[ DiskBox[{2.017540427513767, 0.000442083090181955}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$1"], TagBox[ DiskBox[{2.017313637850549, 0.8021720908379496}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1.0081041532060644`, 0.4011713309635613}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{0., 0.}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{0.00022442849777104534`, 0.8026134053739986}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$5"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{0.9900000000000007, 98.51000000000002}, {-24.144871762947535`, 20.0896}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, ImageSize->{99.5, Automatic}]\)]$

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In[22]:=

$ResourceFunction["PoincarePolynomial"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5}, {Null, SparseArray[ Automatic, {5, 5}, 0, {1, {{0, 2, 4, 8, 10, 12}, {{3}, {2}, {1}, {3}, {2}, { 1}, {4}, {5}, {3}, {5}, {3}, {4}}}, Pattern}]}, {EdgeStyle -> { Arrowheads[Small]}, ImageSize -> {99.5, Automatic}, VertexSize -> {Medium}}]]}, TagBox[GraphicsGroupBox[ GraphicsComplexBox[{{2.017540427513767, 0.000442083090181955}, {2.017313637850549, 0.8021720908379496}, {1.0081041532060644`, 0.4011713309635613}, {0., 0.}, {0.00022442849777104534`, 0.8026134053739986}}, { {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Small], LineBox[{{1, 3}, {1, 2}, {2, 3}, {3, 4}, {3, 5}, {4, 5}}]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.08017300398243705], DiskBox[2, 0.08017300398243705], DiskBox[3, 0.08017300398243705], DiskBox[4, 0.08017300398243705], DiskBox[5, 0.08017300398243705]}}]], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{99.5, Automatic}]\)]$

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This means that the first coefficient corresponds to the number of WeaklyConnectedGraphComponents instead of ConnectedGraphComponents:

In[23]:=

$Length[ConnectedGraphComponents[\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5}, {{{1, 2}, {2, 3}, {3, 1}, {3, 4}, {4, 5}, {3, 5}}, Null}, {EdgeStyle -> { Arrowheads[Small]}, VertexSize -> {Medium}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Arrowheads[0.03374070617041407], Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$1", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], TagBox[ DiskBox[{2.017540427513767, 0.000442083090181955}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$1"], TagBox[ DiskBox[{2.017313637850549, 0.8021720908379496}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1.0081041532060644`, 0.4011713309635613}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{0., 0.}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{0.00022442849777104534`, 0.8026134053739986}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$5"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{0.9900000000000007, 98.51000000000002}, {-24.144871762947535`, 20.0896}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, ImageSize->{99.5, Automatic}]\)]]$

Out[23]=

In[24]:=

$Length[WeaklyConnectedGraphComponents[\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5}, {{{1, 2}, {2, 3}, {3, 1}, {3, 4}, {4, 5}, {3, 5}}, Null}, {EdgeStyle -> { Arrowheads[Small]}, VertexSize -> {Medium}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Arrowheads[0.03374070617041407], Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$1", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[Small], StripOnInput -> False]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], TagBox[ DiskBox[{2.017540427513767, 0.000442083090181955}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$1"], TagBox[ DiskBox[{2.017313637850549, 0.8021720908379496}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1.0081041532060644`, 0.4011713309635613}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{0., 0.}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{0.00022442849777104534`, 0.8026134053739986}, 0.08017300398243705], "DynamicName", BoxID -> "VertexID$5"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{0.9900000000000007, 98.51000000000002}, {-24.144871762947535`, 20.0896}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, ImageSize->{99.5, Automatic}]\)]]$

Out[24]=

The Poincaré polynomial does not identify links, knots or braids:

In[25]:=

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Out[25]=

In[26]:=

Out[26]=

The Poincaré polynomial cannot determine orientability:

In[27]:=

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Out[27]=

In[28]:=

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