(* 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[ 1091484, 18872] NotebookOptionsPosition[ 1087086, 18723] NotebookOutlinePosition[ 1087500, 18741] CellTagsIndexPosition[ 1087457, 18738] 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 One Parameter Knot Theory Computations", "Title", CellChangeTimes->{{3.390968043671875*^9, 3.39096806046875*^9}, 3.390968404171875*^9, {3.391085838140625*^9, 3.3910858503125*^9}}], Cell["By Dror Bar - Natan and Thomas Fiedler", "Text", CellChangeTimes->{{3.391085917921875*^9, 3.391085934265625*^9}}, FontWeight->"Bold"], Cell[TextData[{ "This ", Cell[BoxData[ FormBox[ ButtonBox[ RowBox[{"Mathematica", " ", "notebook"}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://www.math.toronto.edu/~drorbn/Misc/OneParameter/OneParameter.nb"]\ , None}], TraditionalForm]]], " contains some programs that we wrote in June 2007 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}], ".\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 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 Jones polynomial, and while we could not compute enough of the second to \ fully identify it, we computed enough to tell that contrary 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 Jones 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, 2007. It is not easily readable and not \ meant for readership much beyond its own authors. 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