(************** 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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The brilliant terminology is due to Ken Shan.\ \>", "Text"], Cell[BoxData[{ \(\(MinesweeperMatrix[pgd_PlanarGridDiagram]\ := \ Module[\[IndentingNewLine]{l, \ CurrentRow, \ c1, c2, k, s}, \[IndentingNewLine]l = Length[pgd]; \[IndentingNewLine]CurrentRow\ = \ Table[0, \ {l}]; \[IndentingNewLine]Table[\[IndentingNewLine]{c1, \ \ c2}\ = \ Sort[pgd[\([k]\)]]; \[IndentingNewLine]s = Sign[{\(-1\), 1} . pgd[\([k]\)]]; \ \[IndentingNewLine]Do[\[IndentingNewLine]CurrentRow[\([c]\)] += s, \[IndentingNewLine]{c, \ c1, \ c2 - 1}\[IndentingNewLine]]; \[IndentingNewLine]CurrentRow, \ \[IndentingNewLine]{k, l}\[IndentingNewLine]]\[IndentingNewLine]];\)\), "\ \[IndentingNewLine]", \(\(MinesweeperMatrix[K_]\ := \ MinesweeperMatrix[PlanarGridDiagram[K]];\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((t^MinesweeperMatrix[Knot[3, 1]])\)\ // \ Reverse\)\ // \ MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "1", "1", "1"}, {"t", "t", "t", "1", "1"}, 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pgd[\([k, \ 1]\)] - 1]\ + \ a[k - 1, \ pgd[\([k, \ 1]\)]]\ + \ a[k - 1, \ pgd[\([k, \ 1]\)] - 1]\[IndentingNewLine] + \ a[k, \ pgd[\([k, \ 2]\)]]\ + \ a[k, \ pgd[\([k, \ 2]\)] - 1]\ + \ a[k - 1, \ pgd[\([k, \ 2]\)]]\ + \ a[k - 1, \ pgd[\([k, \ 2]\)] - 1], \[IndentingNewLine]{k, n}\[IndentingNewLine]]\)}], "Input"], Cell[BoxData[ \(\(-5\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(AlexanderDegree[x_State]\ := \ \(AlexanderDegree[x]\ = \ AlexanderDegreeShift\ + \ Sum[\[IndentingNewLine]a[k, \ x[\([k]\)]], \[IndentingNewLine]{k, \ n}\[IndentingNewLine]]\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(AlexanderDegree[State[7, 6, 5, 4, 3, 2, 1]]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x0 = State\ @@ \ RotateLeft[\(-1\) + First\ /@ \ \(List\ @@ \ pgd\)]\ /. \ 0\ \[Rule] \ n\)], "Input"], Cell[BoxData[ \(State[6, 5, 4, 3, 2, 1, 7]\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(MaslovDegree[x0]\ = \ 1 - n;\)\), "\[IndentingNewLine]", \(\(WW[{{i1_, \ i2_}, \ {j1_, \ j2_}}]\ := \ \(WW[{{i1, i2}, \ {j1, j2}}]\ = \ Length[Select[\[IndentingNewLine]First\ /@ \ \((List\ @@ \ pgd)\)[\([Range[i1 + 1, \ i2]]\)], \[IndentingNewLine]\((j1 < # \[LessEqual] j2)\) &\[IndentingNewLine]]]\);\)\), \ "\[IndentingNewLine]", \(\(PP[s_State, \ {{i1_, \ i2_}, \ {j1_, \ j2_}}]\ := \ Module[\[IndentingNewLine]{k, l}, \[IndentingNewLine]Sum[\[IndentingNewLine]l = s[\([k]\)]; \[IndentingNewLine]If[i1 < k < i2, \ 1, \ 1/2]* Which[\[IndentingNewLine]j1 < l < j2, \ 1, \[IndentingNewLine]j1 \[Equal] l\ || \ j2 \[Equal] l, \ 1/2, \[IndentingNewLine]True, \ 0\[IndentingNewLine]], \[IndentingNewLine]{k, i1, i2}\[IndentingNewLine]]\[IndentingNewLine]];\)\), "\ \[IndentingNewLine]", \(\(MaslovDegree[x_State]\ := \ \(MaslovDegree[x]\ \ = \ Module[\[IndentingNewLine]{i1, \ i2, \ j1, \ j2, \ y, \ r, \ s}, \[IndentingNewLine]i1\ = \ 1; \ While[x[\([i1]\)]\ \[Equal] \ x0[\([i1]\)], \ \(++i1\)]; \[IndentingNewLine]j1\ = \ x[\([i1]\)]; \ j2\ = \ x0[\([i1]\)]; \[IndentingNewLine]{{i2}}\ = \ Position[x, \ j2]; \[IndentingNewLine]r\ = \ {Sort[{i1, i2}], \ Sort[{j1, j2}]}; \[IndentingNewLine]s\ = \ Sign[i2 - i1]*Sign[j2 - j1]; \[IndentingNewLine]y = x; \ y[\([i1]\)] = j2; \ y[\([i2]\)] = j1; \[IndentingNewLine]MaslovDegree[y]\ + \ s \((PP[x, \ r]\ + \ PP[y, \ r]\ - \ 2 WW[r])\)\[IndentingNewLine]]\);\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MaslovDegree[State[7, 6, 5, 4, 1, 2, 3]]\)], "Input"], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(AllStates[]\ = \ \(\((State\ @@ \ #)\) &\)\ /@ \ Permutations[Range[n]];\)\), "\[IndentingNewLine]", \(\(AllStates[m_, \ a_]\ := \ Select[AllStates[], \ \((MaslovDegree[#] \[Equal] m\ && \ AlexanderDegree[#] \[Equal] a)\) &];\)\)}], "Input"], Cell["\<\ Here are all states with Maslov Degree-5 and Alexander degree 0:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(AllStates[\(-5\), \ 0]\)], "Input"], Cell[BoxData[ \({State[1, 6, 7, 5, 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