(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 548871, 10779]*) (*NotebookOutlinePosition[ 549607, 10804]*) (* CellTagsIndexPosition[ 549563, 10800]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ ButtonBox["Dror Bar-Natan", ButtonData:>{ URL[ "http://www.math.toronto.edu/~drorbn/"], None}, ButtonStyle->"Hyperlink"], ": ", ButtonBox["Odds, Ends, Unfinished", ButtonData:>{ URL[ "http://www.math.toronto.edu/~drorbn/Misc"], None}, ButtonStyle->"Hyperlink"], ":" }], "Text"], Cell[CellGroupData[{ Cell[TextData[ButtonBox["The Manolescu-Ozsvath-Sarkar Complex", ButtonData:>{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/MOSComplex/"], None}, ButtonStyle->"Hyperlink"]], "Section"], Cell[TextData[{ "This Mathematica notebook contains and lightly documents a program I wrote \ in the summer of 2006. The program sets up the Manolescu-Ozsvath-Sarkar (MOS) \ complex and computes its homology, the \"Knot Floer Homology\". The comploex \ itself is described in detail in the article ", ButtonBox["arXiv:math.GT/0607691", ButtonData:>{ URL[ "http://front.math.ucdavis.edu/math.GT/0607691"], None}, ButtonStyle->"Hyperlink"], " by Manolescu, Ozsvath and Sarkar.\n\nThe program is, as expected, a \ failure. The complexity of the MOS complex grows like n!, so only the Knot \ Floer homology of rather small knots can be computed. But thanks to the work \ of many other authors, the Knot Floer homology is known for all knots with up \ to 10 crossings and many bigger ones as well. So no new computations are \ available here. Yet the program is not pointless: I learned by writing it, \ you may learn by reading and using it, and parts of it are recyclable.\n\nFor \ a little while I plan to maintain, upgrade and update this notebook. Let T3 \ be the time now, let T2 be the time you downloaded this file and let T1 be \ the time defined below, in the first line of code. If (T3-T2)>(T2-T1), you \ should download a new copy from ", Cell[BoxData[ FormBox[ ButtonBox[\(\(\(http\)\(:\)\) // \(\(\(www . math . toronto . edu/\(\(~\)\(drorbn\)\)\)/Misc\)/MOSComplex\)/ MOSComplex . nb\), ButtonData:>{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/MOSComplex/MOSComplex.\ nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " (or web-search for the current location)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(T1 = Date[]\)], "Input"], Cell[BoxData[ \({2006, 8, 5, 8, 40, 49.9473837`9.451087599836542}\)], "Output"] }, Open ]], Cell["\<\ T0, the time stamp on the first version of this document, was \ {2006,8,1,20,41,33.442125}\ \>", "Text"], Cell[CellGroupData[{ Cell["Startup", "Subsection"], Cell["\<\ To start, download the package KnotTheory` from The Knot Atlas site, save it \ on your system, change the path information as suits your system, and start \ executing this notebook.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(AppendTo[$Path, \ \ "\"];\)\), "\[IndentingNewLine]", \(<< \ KnotTheory`\)}], "Input"], Cell[BoxData[ \("Loading KnotTheory` version of July 31, 2006, 9:34:35.6875.\nRead more \ at http://katlas.math.toronto.edu/wiki/KnotTheory."\)], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Producing Planar Grid Diagrams", "Subsection"], Cell["First some definitions:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(InterlacedQ[{a_, b_}, \ {c_, d_}]\ := \ \((Signature[{a, b}] Signature[{c, d}] Signature[{a, b, c, d}] === \(-1\))\);\)\), "\[IndentingNewLine]", \(\(Slidable[a_, b_, m_List]\ := \ Module[\[IndentingNewLine]{h}, \[IndentingNewLine]Or[\ \[IndentingNewLine]\(! \((Or\ @@ \ \((\(InterlacedQ[{a, b}, \ #] &\)\ /@ \ m)\))\)\), \[IndentingNewLine]SameQ[ 0, \[IndentingNewLine]Total[ h\ /@ \ Select[\[IndentingNewLine]Sort[ Flatten[ m]], \[IndentingNewLine]\((Min[a, b] < # < Max[a, b])\) &\[IndentingNewLine]]]\ /. \ 2 h[_]\ \[Rule] \ 0\[IndentingNewLine]]\[IndentingNewLine]]\[IndentingNewLine]\ 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