(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 323683, 8483]*) (*NotebookOutlinePosition[ 324453, 8510]*) (* CellTagsIndexPosition[ 324409, 8506]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Planar Enumerator", "Title"], Cell["\tby Stephen Green", "Subsubtitle"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Introduction" }], "Section"], Cell[TextData[{ "The Planar Enumerator is designed to enumerate objects that have a planar \ presentation, such as knots, links, tangles, and 2D surfaces in ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^4\)]], ". It does this by first generating all the planar diagrams of a given \ complexity (the number of crossings for knots) and then finding out which of \ them are isomorphic to each other by finding relations between them \ (Reidemeister moves for knots). The final result is hopefully just one \ diagram from each isomorphism class.\n\nAt this point, however, the program \ is fairly slow and requires a lot of memory to run for higher complexities. \ It still accomplishes its task though." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Installation" }], "Section"], Cell[TextData[{ "The Planar Enumerator is written primarily in C, but requires ", StyleBox["Mathematica", FontSlant->"Italic"], " as well. The front end is done in ", StyleBox["Mathematica", FontSlant->"Italic"], ".\n\nTo install, obtain the tar archive \"planarenumerator.tar.gz\" and \ extract it to some directory. You then need to put the files \"mathlink.h\" \ and \"libML.a\" from your ", StyleBox["Mathematica", FontSlant->"Italic"], " installation into this directory. For my setup, they can be found in the \ directory \ \"/usr/local/Wolfram/Mathematica/4.2/AddOns/MathLink/DeveloperKit/Linux/\ CompilerAdditions\", but that could certainly be different for your \ installation.\n\nOnce you have the files, simply type \"make\" at the command \ prompt, and the Planar Enumerator should compile. You may want to edit the \ Makefile first to optimize it for your system. Also, depending on the speed \ and memory of your computer, you might want to increase \"MAXPDS\", found in \ the file \"planarenumerator.h\". That variable decides how much memory to \ allocate the for the planar diagrams. I have it set to 2000000 diagrams by \ default because my computer can't handle any more.\n\nThe following command \ loads the Planar Enumerator into ", StyleBox["Mathematica", FontSlant->"Italic"], ", as long as the file \"PlanarEnumerator.m\" is in your path." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(<< PlanarEnumerator`\)], "Input"], Cell[BoxData[ \("Planar Enumerator loaded; Stephen Green, NSERC Undergraduate Student \ Research Award 2003 (supervised by Dror Bar-Natan)."\)], "Print"] }, Open ]], Cell["\<\ Minor changes may be needed for your particular system in the \ Makefile or \"PlanarEnumerator.m\". I have it configured for Linux, but if \ you wanted to run it in Windows, for example, you would need to change the \ name of the C executable from \"pe\" to \"pe.exe\".\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " \"PD notation\"" }], "Section"], Cell["\<\ This section explains the notation used by the Planar Enumerator to \ represent planar diagrams. We will use the example of knots. The following represents the trefoil:\ \>", "Text"], Cell[BoxData[ \(\(PD[X[1, 2, 3, 4], X[2, 1, 5, 6], X[4, 3, 6, 5]];\)\)], "Input"], Cell[TextData[{ "Each ", Cell[BoxData[ \(TraditionalForm\`X\)]], " represents one of the vertices. The numbers number the edges. The first \ vertex ", Cell[BoxData[ \(TraditionalForm\`X[1, 2, 3, 4]\)]], " represents the bottom left vertex in the diagram below. 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But for this example of knots, we are \ using ", Cell[BoxData[ \(TraditionalForm\`X\)]], "'s.\n\nFor each type of vertex you must specify three additional numbers: \ the complexity, the degree, and the symmetry." }], "Text"], Cell[CellGroupData[{ Cell["Complexity", "Subsection"], Cell[TextData[{ "The complexity for a knot vertex is the number of crossings that it \ represents. So for an ", Cell[BoxData[ \(TraditionalForm\`X\)]], " vertex, the complexity is 1. Most vertices will have complexity 1. It's \ when you come to relations, that complexities can be higher." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Degree", "Subsection"], Cell[TextData[{ "The degree of a vertex is the number of edges that go out from it. It's \ the same as the valency. For an ", Cell[BoxData[ \(TraditionalForm\`X\)]], " vertex the degree is 4." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Symmetry", "Subsection"], Cell[TextData[{ "The symmetry of a vertex is a number that specifies how many times the \ vertex's list of numbers must be rotated before arriving at an equivalent \ vertex. The symmetry must divide the degree. For an ", Cell[BoxData[ \(TraditionalForm\`X\)]], " vertex, the symmetry is 2. This means that both ", Cell[BoxData[ \(TraditionalForm\`X[1, 2, 3, 4]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`X[3, 4, 1, 2]\)]], " represent the same vertex. However, ", Cell[BoxData[ \(TraditionalForm\`X[2, 3, 4, 1]\)]], " represents a different vertex." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Setting the values", "Subsection"], Cell["\<\ To set the complexity, degree, and symmetry input the \ following:\ \>", "Text"], Cell[BoxData[ \(Complexity[X] = 1; Deg[X] = 4; Symmetry[X] = 2;\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The 'p' vertex", "Subsection"], Cell[TextData[{ "There is a built-in vertex type called ", Cell[BoxData[ \(TraditionalForm\`p\)]], ", for 'point'. It has complexity 0, degree 2, and symmetry 1. It is used \ to connect together two edges that have different edge numbers, and the \ Planar Enumerator will often eliminate it from your planar diagram and \ renumber the edges." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Canonical Form" }], "Section"], Cell["\<\ The function CanonicalForm[] takes a planar diagram in 'PD \ notation' as input and it returns a canonical form of that planar diagram, \ where the edges have been renumbered and the vertices have been reordered. \ The purpose is to be able to tell if two PDs are actually the same or not. \ Here are some examples:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[ PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]]]\)], "Input"], Cell[BoxData[ \(PD[X[1, 2, 3, 4], X[2, 1, 5, 6], X[4, 3, 6, 5]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[ PD[X[1, 2, 3, 4], X[2, 1, 5, 6], X[4, 3, 6, 5]]]\)], "Input"], Cell[BoxData[ \(PD[X[1, 2, 3, 4], X[2, 1, 5, 6], X[4, 3, 6, 5]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[ PD[X[1, 5, 6, 2], X[6, 5, 4, 3], X[1, 2, 3, 4]]]\)], "Input"], Cell[BoxData[ \(PD[X[1, 2, 3, 4], X[1, 5, 6, 2], X[4, 3, 6, 5]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[ PD[X[1, 2, 3, 4], X[1, 6, 7, 2], X[7, 6, 5, 8], X[3, 4, 5, 8]]]\)], "Input"], Cell[BoxData[ \(PD[X[1, 2, 3, 4], X[1, 5, 6, 2], X[3, 4, 7, 8], X[6, 5, 7, 8]]\)], "Output"] }, Open ]], Cell[TextData[{ "Notice that CanonicalForm[] removes ", Cell[BoxData[ \(TraditionalForm\`p\)]], " vertices from PDs:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[PD[X[1, 1, 2, 3], p[2, 3]]]\)], "Input"], Cell[BoxData[ \(PD[X[1, 1, 2, 2]]\)], "Output"] }, Open ]], Cell[TextData[{ "Lets add ", Cell[BoxData[ \(TraditionalForm\`Y\)]], " vertices to our types:" }], "Text"], Cell[BoxData[ \(Complexity[Y] = 1; Deg[Y] = 3; Symmetry[Y] = 1;\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[ PD[Y[1, 4, 100], Y[4, 2, 10], Y[115, 10, 15], Y[15, 14, 12], Y[1, 22, 3], Y[17, 12, 22], Y[100, 115, 17], Y[3, 14, 2]]]\)], "Input"], Cell[BoxData[ \(PD[Y[1, 2, 3], Y[1, 4, 5], Y[2, 6, 7], Y[3, 8, 9], Y[4, 9, 10], Y[5, 11, 6], Y[7, 12, 8], Y[10, 12, 11]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[ PD[Y[1, 2, 3], Y[1, 4, 5], Y[2, 6, 7], Y[3, 8, 9], Y[4, 9, 10], Y[5, 11, 6], Y[7, 12, 8], Y[10, 12, 11]]]\)], "Input"], Cell[BoxData[ \(PD[Y[1, 2, 3], Y[1, 4, 5], Y[2, 6, 7], Y[3, 8, 9], Y[4, 9, 10], Y[5, 11, 6], Y[7, 12, 8], Y[10, 12, 11]]\)], "Output"] }, Open ]], Cell["The previous PD was a cube.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CanonicalForm[PD[Y[3, 5, 1], Y[2, 4, 3], X[5, 4, 2, 1]]]\)], "Input"], Cell[BoxData[ \(PD[Y[1, 2, 3], X[1, 4, 5, 2], Y[3, 5, 4]]\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Diagrams" }], "Section"], Cell["\<\ The function Diagrams[complexity, vertices] produces all the \ non-isomorphic planar diagrams of a given total complexity and with vertices \ of a given type(s). They are all returned in canonical form.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Diagrams[2, \ 2 X]\)], "Input"], Cell[BoxData[ \({PD[X[1, 2, 3, 4], X[1, 4, 3, 2]], PD[X[1, 2, 3, 4], X[2, 1, 4, 3]], PD[X[1, 1, 2, 3], X[2, 3, 4, 4]], PD[X[1, 1, 2, 3], X[3, 4, 4, 2]], PD[X[1, 2, 3, 1], X[3, 4, 4, 2]], PD[X[1, 1, 2, 3], X[2, 4, 4, 3]], PD[X[1, 1, 2, 3], X[3, 2, 4, 4]], PD[X[1, 2, 3, 1], X[2, 4, 4, 3]]}\)], "Output"] }, Open ]], Cell[TextData[{ "Instead of specifying ", Cell[BoxData[ \(TraditionalForm\`2 X\)]], ", you can write 'Several X', to allow zero or more ", Cell[BoxData[ \(TraditionalForm\`X\)]], "'s, such that the total complexity is still the first argument to \ Diagrams[]." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Diagrams[2, \ Several\ X]\)], "Input"], Cell[BoxData[ \({PD[X[1, 2, 3, 4], X[1, 4, 3, 2]], PD[X[1, 2, 3, 4], X[2, 1, 4, 3]], PD[X[1, 1, 2, 3], X[2, 3, 4, 4]], PD[X[1, 1, 2, 3], X[3, 4, 4, 2]], PD[X[1, 2, 3, 1], X[3, 4, 4, 2]], PD[X[1, 1, 2, 3], X[2, 4, 4, 3]], PD[X[1, 1, 2, 3], X[3, 2, 4, 4]], PD[X[1, 2, 3, 1], X[2, 4, 4, 3]]}\)], "Output"] }, Open ]], Cell["\<\ 'Several' is more useful when you want to combine 2 or more types \ of vertices, with some total complexity.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Diagrams[2, Several\ X\ + \ Several\ Y]\)], "Input"], Cell[BoxData[ \({PD[Y[1, 2, 3], Y[1, 3, 2]], PD[Y[1, 1, 2], Y[2, 3, 3]], PD[X[1, 2, 3, 4], X[1, 4, 3, 2]], PD[X[1, 2, 3, 4], X[2, 1, 4, 3]], PD[X[1, 1, 2, 3], X[2, 3, 4, 4]], PD[X[1, 1, 2, 3], X[3, 4, 4, 2]], PD[X[1, 2, 3, 1], X[3, 4, 4, 2]], PD[X[1, 1, 2, 3], X[2, 4, 4, 3]], PD[X[1, 1, 2, 3], X[3, 2, 4, 4]], PD[X[1, 2, 3, 1], X[2, 4, 4, 3]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Loops", "Subsection"], Cell[TextData[{ "There is the option to turn off loops for any vertex type. Just set ", Cell[BoxData[ \(TraditionalForm\`Loops[X] = 0\)]], " to turn them off, and ", Cell[BoxData[ \(TraditionalForm\`Loops[X] = 1\)]], " to turn them back on. By default they are on. Turning them off saves a \ lot of time and greatly reduces the number of PDs produced." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Loops[X] = 0;\)\), "\[IndentingNewLine]", \(Diagrams[2, 2 X]\)}], "Input"], Cell[BoxData[ \({PD[X[1, 2, 3, 4], X[1, 4, 3, 2]], PD[X[1, 2, 3, 4], X[2, 1, 4, 3]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Length[Diagrams[5, 5 X]]\)], "Input"], Cell[BoxData[ \(100\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Loops[X] = 1;\)\), "\[IndentingNewLine]", \(Length[Diagrams[5, 5 X]]\)}], "Input"], Cell[BoxData[ \(4740\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Relations" }], "Section"], Cell["\<\ Relations are the moves that are allowed to be performed on parts \ of planar diagrams, and give you planar diagrams that are different from what \ you started with, but that represent the same object. For example, with \ knots and links, the relations are the Reidemeister moves: R1, R2, and R3. 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\(TraditionalForm\`LeftHS[R1[a_, b_], c_]\)]], ". The 'c' is so that the program can give a number to the new edge. The \ second and fourth lines of the statement specify how many new edges each side \ has.\n\nFor each relation, you must also specify the complexity, degree, and \ symmetry as with ordinary vertices. The complexity should be set to the \ complexity of the higher complexity side." }], "Text"], Cell[BoxData[ \(Complexity[R1] = 1; Deg[R1] = 2; Symmetry[R1] = 2;\)], "Input"], Cell[TextData[{ "Here's the other two Reidemeister moves along with their descriptions to \ ", StyleBox["Mathematica", FontSlant->"Italic"], ". 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For knots, R1 and R3 \ both have them. It can be advantageous to use these because they can improve \ the results from the Enumerate[] function discussed in the next section. We shall call these mirror images 'R1P' and 'R3P':\ \>", "Text"], Cell[BoxData[{ \(\(LeftHS[R1P[a_, b_], c_] := X[a, b, c, c];\)\), "\[IndentingNewLine]", \(\(LeftHS[R1P] = 1;\)\), "\[IndentingNewLine]", \(\(RightHS[R1P[a_, b_]] := p[a, b];\)\), "\[IndentingNewLine]", \(\(RightHS[R1P] = 0;\)\), "\[IndentingNewLine]", \(Complexity[R1P] = 1; \ Deg[R1P] = 2; \ Symmetry[R1P] = 2; LeftHS[R3P[a_, b_, c_, d_, e_, f_], g_, h_, i_] := PD[X[b, i, g, a], X[c, d, h, i], X[g, h, e, f]];\), "\[IndentingNewLine]", \(\(LeftHS[R3P] = 3;\)\), "\[IndentingNewLine]", \(\(RightHS[R3P[a_, b_, c_, d_, e_, f_], g_, h_, i_] := PD[X[h, i, f, a], X[b, c, g, h], X[g, d, e, i]];\)\), "\[IndentingNewLine]", \(\(RightHS[R3P] = 3;\)\), "\[IndentingNewLine]", \(Complexity[R3P] = 3; Deg[R3P] = 6; Symmetry[R3P] = 6;\)}], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Enumerate" }], "Section"], Cell["\<\ Enumerate[complexity, complexitybound, vertices, relations] will \ find all of the PDs made up of 'vertices' with complexity 'complexity' that \ cannot be reduced to a lower complexity by any of the 'relations'. Also, if \ more than one of the PDs can be linked through relations (going up to PDs \ with complexity 'complexitybound') to other PDs, then only one PD from the \ list of linked PDs is returned. Essentially, Enumerate[] attempts to find all the PDs that represent objects \ that are not isomorphic to eachother. By specifying a higher \ 'complexitybound', results will be improved, but more computer resources will \ be required.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Enumerate[2, 2, Several\ X, \ {R1, R1P, R2, R3, R3P}]\)], "Input"], Cell[BoxData[ \("Generating PDs"\)], "Print"], Cell[BoxData[ \("Setting up lists in C"\)], "Print"], Cell[BoxData[ \("Finding Eliminateable PDs"\)], "Print"], Cell[BoxData[ InterpretationBox[\(\(R1 + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]2\), SequenceForm[ Plus[ R1, Times[ Several, X]], ": complexity = ", 2], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(\(R1P + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]2\), SequenceForm[ Plus[ R1P, Times[ Several, X]], ": complexity = ", 2], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(\(R2 + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]2\), SequenceForm[ Plus[ R2, Times[ Several, X]], ": complexity = ", 2], Editable->False]], "Print"], Cell[BoxData[ \("Finding Relations"\)], "Print"], Cell[BoxData[ \({PD[X[1, 2, 3, 4], X[2, 1, 4, 3]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(<< KnotTheory`\)], "Input"], Cell[BoxData[ \("Loading KnotTheory`..."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[DrawPD[PD[X[1, 2, 3, 4], X[2, 1, 4, 3]]]];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 1.09081 0.5 1.17385 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash newpath matrix currentmatrix 0.306851 0.330208 scale 2.18132 1.95628 1 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In this case, 'complexitybound' did not need to be increased.\n\ \nNote that the relations must be enclosed in a List for the program to work.\ \n\nEnumerate[] also prints out the status of what task it is performing, \ while the C program prints to a stderr window information such as how many \ PDs are produced.\n\nWe drew the result using Emily Redelmeier's program at \ ", ButtonBox["http://www.math.toronto.edu/~drorbn/KAtlas/Manual/DrawPD.html.", ButtonData:>{ URL[ "http://www.math.toronto.edu/~drorbn/KAtlas/Manual/DrawPD.html"], None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Enumerate[3, 3, Several\ X, \ {R1, R1P, R2, R3, R3P}]\)], "Input"], Cell[BoxData[ \("Generating PDs"\)], "Print"], Cell[BoxData[ \("Setting up lists in C"\)], "Print"], Cell[BoxData[ \("Finding Eliminateable PDs"\)], "Print"], Cell[BoxData[ InterpretationBox[\(\(R1 + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]3\), SequenceForm[ Plus[ R1, Times[ Several, 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}, Open ]], Cell["\<\ With complexity 3, we get the two mirror images of the trefoil. Here we see the importance of using 'Several' instead of specific numbers in \ describing the vertices for the Enumerate[] function. As you can see from \ the printout, it generated PDs with 'R1 + Several X' vertices and 'R3 + \ Several X' vertices, both with complexity 3. R1 and R3 have different \ complexities themselves, so really 'R1 + Several X' was like calling 'R1 + \ 2X', and 'R3 + Several X' was like calling 'R3'. So if a specific number was \ put in place of 'Several', the program wouldn't work.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Enumerate[4, 4, Several\ X, \ {R1, R1P, R2, R3, R3P}]\)], "Input"], Cell[BoxData[ \("Generating PDs"\)], "Print"], Cell[BoxData[ \("Setting up lists in C"\)], "Print"], Cell[BoxData[ \("Finding Eliminateable PDs"\)], "Print"], Cell[BoxData[ InterpretationBox[\(\(R1 + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]4\), SequenceForm[ Plus[ R1, Times[ Several, X]], ": complexity = ", 4], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(\(R1P + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]4\), SequenceForm[ Plus[ R1P, Times[ Several, X]], ": complexity = ", 4], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(\(R2 + Several\ X\)\[InvisibleSpace]": complexity = "\[InvisibleSpace]4\), SequenceForm[ Plus[ R2, Times[ Several, X]], ": complexity = ", 4], Editable->False]], "Print"], Cell[BoxData[ 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{{408.875, \ 528.312}, {122.438, 2.9375}} -> {-4.64788, -0.569934, 0.00993757, 0.0104353}}] }, Open ]], Cell["\<\ Now, all these 4 results represent different knots or links. However, it \ took much longer to run.\ \>", "Text"], Cell[CellGroupData[{ Cell["Loops", "Subsection"], Cell["\<\ Something that might be worth looking into is turning off loops before \ calling Enumerate[]. This would mean that not as many PDs would be produced \ at each level, saving time. However, not as many relations would be found \ between PDs, requiring a higher 'complexitybound' to achieve the same \ results.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " How The Planar Enumerator Works" }], "Section"], Cell["\<\ In this section, a rough outline of how the program works is given. For more \ details, consult the comments in the code.\ \>", "Text"], Cell[CellGroupData[{ Cell["Diagrams", "Subsection"], Cell[TextData[{ "The Diagrams[] function is written mainly in C, but the C function is \ called by a ", StyleBox["Mathematica", FontSlant->"Italic"], " function that formats the input a bit. Most of the code for the \ Diagrams[] function is found in the file \"PDgenerator.c\".\n\nFirst, all of \ the adjacency matrices that could represent diagrams with some degree \ sequence are produced. In each entry of an adjacency matrix we find the \ number of edges between the vertex represented by the column and the vertex \ represented by the row. For example for the Hopf Link, we would get the \ adjacency matrix ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"0", "4"}, {"4", "0"} }], ")"}], TraditionalForm]]], ", since each vertex makes 4 connections with the other vertex. If you \ swap row i with row j and column i with column j in an adjacency matrix, you \ obtain a new adjacency matrix which represents the same object. A few checks \ are done to try to reduce the number of isomorphic adjacency matrices \ produced, but still, many isomorphic ones are produced at the higher \ complexities. Finding a canonical form for adjacency matrices would speed up \ the program greatly.\n\nTurning loops on or off is implented as the adjacency \ matrices are generated, by allowing or disallowing non-zero entries along the \ diagonals of the matrices.\n\nNext, from each adjacency matrix, we produce \ all the diagrams (planar and non-planar) that the matrix could represent. At \ this stage, we treat all vertices of the same degree as equivalent, and \ assign each vertex symmetry 1. We then filter out the non-planar ones by \ using Euler's Formula which says that:\n\n", Cell[BoxData[ \(TraditionalForm\`\((number\ of\ faces)\)\ - \ \((number\ of\ \ edges)\)\ + \ \((number\ of\ vertices)\)\ = \ 2\)]], "\n\nThe planar ones are put into canonical form, and the duplicates \ removed. These planar diagrams are called 'basic PDs' since all vertices of \ the same degree are equivalent, and symmetry is always 1.\n\nFor each basic \ PD, we then assign types to each of its vertices and rotate their edge lists \ according to their symmetry in order to obtain all the PDs it could \ represent. Each PD is put into canonical form and duplicates are again \ removed. When this is done for every basic PD, we have the final result.\n\n\ With this technique, we generate all the planar and non-planar graphs, even \ though we never use the non-planar graphs. Since the non-planar graphs \ greatly outnumber the planar ones, it would be worthwhile to find a new \ method that somehow only finds the planar graphs." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Enumerate", "Subsection"], Cell[TextData[{ "Enumerate[] is also primarily implemented in C, but the C function makes \ calls to ", StyleBox["Mathematica", FontSlant->"Italic"], " through ", StyleBox["MathLink", FontSlant->"Italic"], " to access the RightHS[] and LeftHS[] functions that the user must define \ for the relations. Also, the ", StyleBox["Mathematica", FontSlant->"Italic"], " function Enumerate[] calls the C functions to get things started. Most \ of the C code is found in the file \"PDhashtable.c\". Also, this makes use \ of a hash table available at ", ButtonBox["http://burtleburtle.net/bob/hash/hashtab.html", ButtonData:>{ URL[ "http://burtleburtle.net/bob/hash/hashtab.html"], None}, ButtonStyle->"Hyperlink"], ".\n\nFirst, all of the PDs with the desired vertices with complexities \ ranging from 'complexity' to 'complexitybound' are produced. Each one is \ then put into a hash table as the key, with an associated value which \ consists of a number. These numbers are just assigned sequentially starting \ with 1.\n\nThen the program generates two int arrays, each of the same length \ as the total number of PDs already produced. One of them, called \ 'eliminateablePDs' contains all zeros. The other, called 'relations', \ contains an int in each element which is equal to the element number.\n\nThe \ program then finds all the PDs containing a relation which have one side with \ complexity within the range we're looking at and the other side with \ complexity below 'complexity'. The upper side of each of these can then be \ reduced by a single move to a PD of complexity less than we want. We then \ change the value of the element in 'eliminateablePDs' corresponding to the \ upper complexity side to 1 from 0.\n\nWe then find all the PDs containing a \ relation which have both sides within the range we're looking at. We take \ the LeftHS and RightHS of the relation to get the two PDs, which we put into \ canonical form. Then using the hash table we find their indices. To finish \ off, we change the value of the element of 'relations' which corresponds to \ the upper complexity PD to the index of the lower complexity PD. Also, if \ the element of 'eliminateablePDs' corresponding to the upper complexity side \ is 1, then we change the value of 'eliminateablePDs' corresponding to the \ lower complexity side to 1 as well. If the value of the upper complexity \ element in 'relations' is something other than the element number, then it \ already points somewhere else so we do some more changes.\n\nFinally, we \ return all of the PDs of complexity 'complexity' that have 'eliminateablePDs' \ set to 0 and that have 'relations' point to themselves." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Future Improvements" }], "Section"], Cell[TextData[{ "At this point, the program is too slow and requires too much memory. The \ number 7 seems to be the limit for the complexity with knots (loops turned \ on) on my machine (Athlon 1.2 GHz with 512 MB of RAM) due to the memory \ usage. The number of PDs produced increases very rapidly as the complexity \ increases, so storing them all becomes a problem. Solutions to the memory \ problem could include using some sort of disk-based database instead of that \ hash table stored in memory, and finding a more compact way of storing the \ PDs.\n\nAlthough the memory is the first bottleneck I encounter, if this can \ be fixed, speed will become an issue. In the generation of PDs, a large \ improvement could be found if we either found a canonical form for the \ adjacency matrices, or found a way to generate only the planar graphs, as \ discussed above. In the enumeration part of the program, a lot of speed was \ gained by moving away from ", StyleBox["Mathematica", FontSlant->"Italic"], " into C (the ", StyleBox["Mathematica", FontSlant->"Italic"], " Dispatch tables were way to slow), but we still must make many calls \ through ", StyleBox["MathLink", FontSlant->"Italic"], " in order to access the LeftHS and RightHS functions defined by the user. \ If a method was found to move these functions into C and still make them easy \ for the user to define, then things would be greatly sped up. However, the \ first priority when it comes to speed is probably improving the planar \ diagram generation.\n\nA new feature that would be fairly easy to implement \ is an option in the Enumerate function to apply a filter to all the PDs as \ they are generated, before they are put into the hash table. This could be \ used to filter out knots from all the diagrams with ", Cell[BoxData[ \(TraditionalForm\`X\)]], " vertices. Later on, functions to \"relax\" diagrams and tell if two \ diagrams represent the same object or not, with a list of moves to show how \ they are the same would be useful." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.2 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{957, 959}, WindowMargins->{{Automatic, 69}, {Automatic, 4}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, CellLabelAutoDelete->True, Magnification->1, StyleDefinitions -> "Default.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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