(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 12611, 519]*) (*NotebookOutlinePosition[ 13320, 543]*) (* CellTagsIndexPosition[ 13276, 539]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Some Khovanov-Rozansky Computations", "Title", FontSize->24], Cell["\<\ See http://www.math.toronto.edu/~drorbn/Misc/KhovanovRozansky/\ \>", "Subtitle", FontSize->16], Cell["\<\ My programs are admittedly pathetic; they were written mostly to verify that \ I understand the definitions correctly. They are inelegant and inefficient \ and they hardly compute anything. Even the trefoil knot seems too large for \ these programs. (Though what's computed is enough to convince me that I do \ understand the definitions, and may already be of some very minor interest to \ others). Nothing was checked too carefully and all may be wrong.\ \>", "Text"], Cell[CellGroupData[{ Cell["Startup:", "Subsection"], Cell["\<\ We first have to load some programs: the package KnotTheory`, and the files \ SubQuotient.m and KRH.m. If you are wise and they are all in the same \ directory, just load all three. On my \"development\" machine, where each one \ is saved in a different place, the following sequence is necessary:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(SetDirectory["\"];\)\), "\ \[IndentingNewLine]", \(\(AppendTo[$Path, \ "\"];\)\), "\ \[IndentingNewLine]", \(\(Get["\"];\)\), "\[IndentingNewLine]", \(\(Get["\<../SubQuotient/SubQuotient.m\>"];\)\), "\[IndentingNewLine]", \(\(Get["\"];\)\), "\[IndentingNewLine]", \(\(n = 2;\)\)}], "Input"], Cell[BoxData[ \("Loading KnotTheory`..."\)], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Here are some n=2 computations for the Hopf link, Link[2, Alternating, 1]:\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(K = PD[Link[2, Alternating, 1]];\)\)], "Input"], Cell[BoxData[ \(KnotTheory::"loading" \(\(:\)\(\ \)\) "Loading precomputed data in \!\(\"PD4Links`\"\)."\)], "Message"] }, Open ]], Cell["Degree -8, height 0 through 2:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Dim[KRH[K, \ #, \ 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2]]\ &\)\ /@ \ {0, 1, 2}\)], "Input"], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ (These numbers agree with the ordinary Khovanov homology for the Hopf link, \ as expected for n=2)\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Here are some n=2 computations for the trefoil knot, Knot[3,1]:\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(n = 2; \ K = PD[Knot[3, 1]];\)], "Input"], Cell[BoxData[ \(KnotTheory::"loading" \(\(:\)\(\ \)\) "Loading precomputed data in \!\(\"PD4Knots`\"\)."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Dim[KRH[K, \ #, \ \(-11\)]]\ &\)\ /@ \ {0, 1, 2, 3}\)], "Input"], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Dim[KRH[K, \ #, \ \(-9\)]]\ &\)\ /@ \ {0, 1, 2, 3}\)], "Input"], Cell[BoxData[ \({0, 0, 0, 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Dim[KRH[K, \ #, \ \(-7\)]]\ &\)\ /@ \ {0, 1, 2, 3}\)], "Input"], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Dim[KRH[K, \ #, \ \(-5\)]]\ &\)\ /@ \ {0, 1, 2, 3}\)], "Input"], Cell[BoxData[ \({0, 0, 1, 0}\)], "Output"] }, Open ]], Cell["\<\ (These numbers agree with the ordinary Khovanov homology for the trefoil \ knot, as expected for n=2)\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Here are some n=3 computations for the Hopf link, Link[2, Alternating, 1]:\ \>", "Subsection"], Cell["The easiest way to change n is to restart mathematica:", "Text"], Cell[BoxData[ \(Exit[]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(SetDirectory["\"];\)\), "\ \[IndentingNewLine]", \(\(AppendTo[$Path, \ "\"];\)\), "\ \[IndentingNewLine]", \(\(Get["\"];\)\), "\[IndentingNewLine]", \(\(Get["\<../SubQuotient/SubQuotient.m\>"];\)\), "\[IndentingNewLine]", \(\(Get["\"];\)\), "\[IndentingNewLine]", \(\(n = 3;\)\)}], "Input"], Cell[BoxData[ \("Loading KnotTheory`..."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ 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