(************** 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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