(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 1202025, 25496] NotebookOptionsPosition[ 1195897, 25295] NotebookOutlinePosition[ 1196376, 25315] CellTagsIndexPosition[ 1196333, 25312] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ ButtonBox["Dror Bar-Natan", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.math.toronto.edu/~drorbn/"], None}], ": ", ButtonBox["Odds, Ends, Unfinished", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.math.toronto.edu/~drorbn/Misc"], None}], ":" }], "Text"], Cell[CellGroupData[{ Cell["Some HOMFLY-PT One Parameter Knot Theory Computations", "Title", CellChangeTimes->{{3.390968043671875*^9, 3.39096806046875*^9}, 3.390968404171875*^9, {3.391085838140625*^9, 3.3910858503125*^9}, { 3.4230640231794*^9, 3.4230640285769997`*^9}}], Cell["\<\ By Jana Archibald, Dror Bar - Natan, Karene Chu and Thomas Fiedler\ \>", "Text", CellChangeTimes->{{3.391085917921875*^9, 3.391085934265625*^9}, { 3.4230640410726*^9, 3.4230640627566*^9}}, FontWeight->"Bold"], Cell[TextData[{ "This ", Cell[BoxData[ FormBox[ ButtonBox[ ButtonBox[ ButtonBox[ RowBox[{"Mathematica", " ", "notebook"}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/OneParameter/OneParameter.\ nb"], None}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://katlas.math.toronto.edu/drorbn/MathBlog/2008-06/OneParameterY.\ nb"], None}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/OneParameter/OneParameterY.\ nb"], None}], TraditionalForm]]], " contains some programs that we wrote in June 2008 to compute some knot \ invariants coming out of Fiedler's \"one-parameter\" approach to knot theory \ as described in ", ButtonBox["arXiv:math/0612115", BaseStyle->"Hyperlink", ButtonData->{ URL["http://arxiv.org/abs/math/0612115"], None}], ". This is much like the computations by Bar-Natan and Fiedler from ", ButtonBox["June 2007", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.math.toronto.edu/~drorbn/Misc/OneParameter/index.html"], None}], ", except this time the underlying knot invariant is the HOMFLY-PT \ polynomial instead of the Jones polynomial.\n\nIn short, the idea is to \ associate in some canonical way a \"knot movie\", or more precisely, a \ \"one-parameter family of knot diagrams\" to a given knot diagram. Within a \ generic knot movie some \"frames\" will be singular. In the said article \ Fiedler constructs a state sum invariant of \"knot diagrams with a triple \ crossing\" in much of the same way as knot invariants are constructed using \ state sums involving just standard knot diagrams. When those \ \"triple-crossing state sums\" satisfy certain complicated conditions, they \ yield invariants of knot movies, and when evaluated on the movies canonicaly \ associated with knot diagrams, they yield knot invariants.\n\nThis time \ around we rely on unpublished work by Fiedler, in which the relevant \"state \ sum\" is replaced by a HOMFLY-PT-inspired invariant.\n\nThis construction \ seems new and little is known about it. While the invariants it produces are \ always finite type, they are otherwise potentially new.\n\nIn this notebook \ we compute two of the simplest knot invariants introduced by Fiedler. The \ results are in some sense disappointing - the first of the two invariants \ quite clearly comes out to by a multiple of the HOMFLY-PT polynomial, and \ while we could not compute enough of the second to fully identify it, we \ computed enough to tell that it does not separate ", Cell[BoxData[ FormBox[ SubscriptBox["8", "17"], TraditionalForm]]], " from its inverse. So no jackpot here, but a new and unexplained \ construction of the HOMFLY-PT polynomial is surely an interesting thing. \ Where does it come from? What exactly is the universe of all such \ constructions? Are there news in any of the others?\n\nThe main purpose of \ this notebook is to preserve our efforts of June 16-22, 2008. It is not \ easily readable and not meant for readership much beyond its own authors. \ Don't bother complaining.\n\nAlong the way we had to compute the HOMFLY-PT \ polynomial for rather large knots, ones fro which the existing code in the \ package ", ButtonBox["KnotTheory`", BaseStyle->"Hyperlink", ButtonData->{ URL["http://katlas.org/wiki/The_Mathematica_Package_KnotTheory"], None}], " was hugely insufficient. Therefore we wrote some new Hecke-algebra \ sparse-matrix code to compute the HOMFLY-PT polynomial (the program \"Homf\", \ below). This code is not yet well tested and is not yet a part of \ KnotTheory`, but we hope to incorporate it into KnotTheory` soon.\n\nThe .nb \ (original) version of this notebook is at ", Cell[BoxData[ FormBox[ ButtonBox[ ButtonBox[ RowBox[{ RowBox[{"http", ":"}], "//", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"www", ".", "math", ".", "toronto", ".", "edu"}], "/", RowBox[{"~", "drorbn"}]}], "/", "Misc"}], "/", "OneParameter"}], "/", RowBox[{"OneParameterY", ".", "nb"}]}]}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/OneParameter/OneParameter.\ nb"], None}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/OneParameter/OneParameterY.\ nb"], None}], TraditionalForm]]], ". Some blackboard shots taken when we were working ar at ", ButtonBox["http://katlas.math.toronto.edu/drorbn/bbs/show?shot=Fiedler-\ 080616-084319.jpg", BaseStyle->"Hyperlink", ButtonData->{ URL["http://katlas.math.toronto.edu/drorbn/bbs/show?shot=Fiedler-080616-\ 084319.jpg"], None}], "." }], "Text", CellChangeTimes->{{3.39108588553125*^9, 3.3910859125625*^9}, { 3.39108596478125*^9, 3.391086049734375*^9}, {3.391086082359375*^9, 3.391086110546875*^9}, {3.3910861784375*^9, 3.391086614390625*^9}, { 3.391086665828125*^9, 3.3910867760625*^9}, {3.391086857265625*^9, 3.391087038828125*^9}, {3.391087076234375*^9, 3.391087360484375*^9}, { 3.39109244975*^9, 3.39109244975*^9}, {3.391092565953125*^9, 3.39109258140625*^9}, {3.391092660375*^9, 3.391092705765625*^9}, { 3.391100621625*^9, 3.391100635546875*^9}, {3.391100667875*^9, 3.391100667890625*^9}, {3.391100727265625*^9, 3.391100741734375*^9}, { 3.4230642904854*^9, 3.4230643953018*^9}, {3.4230644361113997`*^9, 3.4230644361113997`*^9}, 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