(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 147078, 2615]*) (*NotebookOutlinePosition[ 148065, 2648]*) (* CellTagsIndexPosition[ 148021, 2644]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooo 002]oooo0P000oooo0`0008[oool00:Oo ool20000d?ooo`80002:oooo002Voooo0P000=7oool30000ROooo`00YOoo o`<0003Boooo0P0008Woool00:Coool30000doooo`<00028oooo002Toooo 0P000=Goool20000R?ooo`00Xoooo`80003Foooo0P0008Soool00:;oool3 0000eoooo`800027oooo002Qoooo0`000=Soool20000Qoooo`00XOooo`80 003Joooo0P0008Koool00:3oool20000foooo`800026oooo002Ooooo0`00 0=coool20000QOooo`00Woooo`80003Moooo0P0008Goool009koool30000 gOooo`<00024oooo002Noooo0P000=ooool20000Q?ooo`00WOooo`80003P oooo0`0008?oool009goool20000hOooo`800023oooo002Loooo0P000>;o ool30000P_ooo`00Voooo`<0003Soooo0P0008;oool009_oool20000i?oo o`<00021oooo002Joooo0`000>Goool20000POooo`00V_ooo`80003Voooo 0`00083oool009Woool20000j?ooo`800020oooo002Ioooo0P000>Soool3 0000Ooooo`00V?ooo`80003Zoooo0P0007ooool009Ooool30000j_ooo`<0 001noooo002Goooo0P000>coool20000O_ooo`00U_ooo`80003]oooo0`00 07goool009Koool20000k_ooo`80001moooo002Eoooo0P000>ooool20000 OOooo`00U?ooo`<0003`oooo0P0007coool009Coool20000lOooo`80001l oooo002Coooo0`000?7oool20000O?ooo`00Toooo`80003coooo0P0007_o ool009?oool20000loooo`80001koooo002Boooo0P000?Coool20000Nooo o`00T_ooo`80003eoooo0P0007[oool0097oool20000m_ooo`80001joooo 002Aoooo0P000?Koool30000NOooo`00T?ooo`800022oooo100007;oool2 0000NOooo`00T?ooo`800020oooo1`00077oool30000N?ooo`00Soooo`<0 0020oooo0`0000Coool00`000?ooooooo`1_oooo0P0007Soool008ooool2 0000P?ooo`<0001goooo0`0007Ooool008koool30000P?ooo`80001ioooo 0P0007Ooool008koool20000POooo`80001ioooo0`0007Koool008koool2 0000POooo`80001joooo0P0007Koool008goool20000P_ooo`80001joooo 0`0007Goool008goool20000P_ooo`80001koooo0P0007Goool008coool2 0000Q?ooo`030000oooooooo00?oool20000M?ooo`80001eoooo002oooo0022oooo0`0006Coool?0000=_oooa400036oooo0022 oooo0P0005koool?0000AOooo`h00031oooo0022oooo0P0005Ooool?0000 E?ooo`/0002loooo0022oooo0P00053oool?0000HOooo`/0002foooo0022 oooo0P0004[oool>0000KOooo`T0002coooo0021oooo0P0004Coool>0000 NOooo`L0002`oooo0021oooo0P0003ooool<0000Q?ooo`H0002]oooo0021 oooo0P0003Woool<0000SOooo`H0002Zoooo0021oooo0P0003Coool;0000 U_ooo`L0002Voooo0020oooo0`0002ooool;0000W_ooo`H0002Toooo0020 oooo0P0002coool:0000Y_ooo`H0002Qoooo0020oooo0P0002Soool90000 [_ooo`D0002Ooooo0020oooo0P0002;oool;0000]?ooo`D0002Moooo0020 oooo0P0001koool;0000^oooo`D0002Joooo001ooooo0P0001coool80000 `oooo`D0002Hoooo001ooooo0P0001coool40000bOooo`H0002Eoooo001o oooo0P000>coool50000Toooo`00Ooooo`80003^oooo1@00097oool007ko ool30000l?ooo`@0002@oooo001noooo0P000??oool40000S_ooo`00O_oo o`80003eoooo100008coool007koool20000moooo`@0002:oooo001noooo 0P000?Woool40000R?ooo`00OOooo`80003loooo100008Koool007goool2 0000oOooo`D00024oooo001moooo0P000?ooool40000Poooo`00OOooo`80 003ooooo0_ooo`@00021oooo001loooo0`000?ooool4oooo0`00083oool0 07coool20000ooooo`Koool40000O_ooo`00O?ooo`80003ooooo2?ooo`<0 001moooo001loooo0P000?ooool9oooo100007_oool007coool20000oooo o`[oool40000N_ooo`00Noooo`80003ooooo3Oooo`<0001ioooo001koooo 0P000?ooool>oooo100007Ooool007_oool20000oooooa3oool30000M_oo o`00Noooo`80003ooooo4Oooo`@0001doooo001joooo0`000?oooolCoooo 0`0007?oool007[oool20000oooooaGoool30000L_ooo`00N_ooo`80003o oooo5_ooo`@0001`oooo001joooo0P000?oooolHoooo0`0006ooool007[o ool20000oooooaWoool30000K_ooo`00N_ooo`80003ooooo6_ooo`<0001] oooo001joooo0P000?oooolKoooo100006_oool007[oool20000oooooago ool30000J_ooo`00N_ooo`80003ooooo7_ooo`<0001Yoooo001joooo0P00 0?oooolOoooo0`0006Soool007[oool20000ooooob3oool40000I_ooo`00 N_ooo`80003ooooo8_ooo`<0001Uoooo001joooo0P000?oooolSoooo0`00 06Coool007[oool20000ooooobCoool30000Hoooo`00N_ooo`80003ooooo 9Oooo`<0001Roooo001joooo0P000?oooolVoooo0`00067oool007[oool2 0000ooooobOoool30000H?ooo`00N_ooo`80003ooooo:?ooo`80001Poooo 001joooo0P000?oooolYoooo0P0005ooool007Woool30000ooooob[oool2 0000G_ooo`00DOooo`030000oooooooo02Koool20000ooooob_oool20000 GOooo`00D?ooo`<0000Woooo0P000?ooool[oooo0`0005coool004ooool3 0000:?ooo`80003ooooo;?ooo`<0001Koooo001>oooo0`0002Woool20000 ooooobgoool30000F_ooo`00COooo`<0000Zoooo0P000?ooool^oooo0`00 05Woool004goool20000:oooo`80003ooooo;oooo`<0001Hoooo001_ooo`<0001=oooo0012 oooo0`0003Goool30000oooooc[oool20000COooo`00@_ooo`80000goooo 0P000?oooolkoooo0P0004coool0047oool20000>?ooo`80003ooooo>ooo o`<0001;oooo0010oooo0`0003Soool20000oooooaGoool200009Oooo`80 001;oooo000ooooo0`0003Woool20000oooooaGoool200009_ooo`80001: oooo000ooooo0P0003[oool20000oooooaCoool300009_ooo`<00019oooo 000noooo0P0003_oool20000oooooaCoool20000:?ooo`<00018oooo000m oooo0`0003_oool20000oooooaCoool20000:Oooo`800018oooo000loooo 0`0003coool20000oooooaCoool20000:_ooo`800017oooo000loooo0P00 03goool30000oooooa;oool30000:_ooo`<00016oooo000koooo0P0003oo ool20000oooooa;oool20000;?ooo`800016oooo000joooo0`0003ooool2 0000oooooa;oool20000;Oooo`800015oooo000ioooo0`00043oool20000 oooooa7oool30000;Oooo`<00014oooo000ioooo0P00047oool20000oooo oa7oool20000;oooo`800014oooo000hoooo0P0004;oool20000oooooa7o ool20000?ooo`80000moooo000boooo 0P0004[oool20000ooooo`coool20000>Oooo`<0000loooo000aoooo0P00 04_oool20000ooooo`coool20000>_ooo`80000loooo000`oooo0`0004_o ool20000ooooo`_oool20000??ooo`80000koooo000`oooo0P0004coool2 0000ooooo`_oool20000??ooo`80000koooo000_oooo0P0004koool20000 ooooo`Woool30000?Oooo`80000joooo000^oooo0`0004koool20000oooo o`Woool20000?_ooo`<0000ioooo000^oooo0P0004ooool20000ooooo`Wo ool20000?oooo`80000ioooo000]oooo0P00053oool20000ooooo`Soool2 0000@Oooo`80000hoooo000/oooo0`00057oool20000ooooo`Ooool20000 @Oooo`80000hoooo000/oooo0P0005;oool20000ooooo`Koool30000@_oo o`80000goooo000[oooo0P0005?oool20000ooooo`Koool20000@oooo`<0 000foooo000Zoooo0`0005?oool20000ooooo`Koool20000A?ooo`80000f oooo000Zoooo0P0005Coool30000ooooo`Coool20000AOooo`<0000eoooo 000Yoooo0P0005Koool20000ooooo`Coool20000A_ooo`80000eoooo000X oooo0`0005Koool20000ooooo`Coool20000A_ooo`<0000doooo000Xoooo 0P0005Ooool20000ooooo`?oool20000B?ooo`80000doooo000Woooo0P00 05Woool20000ooooo`;oool20000B?ooo`<0000coooo000Voooo0`0005Wo ool20000ooooo`;oool20000BOooo`80000coooo000Voooo0P0005[oool2 0000ooooo`7oool20000B_ooo`<0000boooo000Uoooo0P0005coool20000 ooooo`80001;oooo0P0003;oool002Coool30000G?ooo`80003noooo0`00 04_oool30000ooool20000GOooo`80000Xoooo000Goooo 0P0007;oool30000kOooo`80001Noooo0P0002Soool001Ooool20000Looo o`80003]oooo0P0005koool300009oooo`005_ooo`80001doooo0`000>_o ool30000Goooo`80000Woooo000Foooo0P0007Goool20000joooo`80001P oooo0P0002Ooool001Goool30000M_ooo`80003Yoooo0`00067oool20000 9_ooo`005Oooo`80001goooo0`000>Soool20000H_ooo`80000Voooo000D oooo0`0007Soool20000j?ooo`80001Soooo0P0002Goool001Coool20000 NOooo`<0003Voooo0P0006Coool200009Oooo`004oooo`<0001joooo0P00 0>Koool20000I?ooo`<0000Toooo000Coooo0P0007_oool30000i?ooo`80 001Voooo0P0002Coool001;oool30000O?ooo`80003Toooo0P0006Koool2 00009?ooo`004_ooo`80001noooo0P000>;oool20000Ioooo`80000Toooo 000Aoooo0`0007koool30000hOooo`80001Woooo0`0002?oool0017oool2 0000P?ooo`80003Poooo0`0006Soool200008oooo`004?ooo`<00020oooo 0`000=ooool20000JOooo`80000Soooo000@oooo0P0008;oool20000g_oo o`<0001Yoooo0P0002?oool000ooool30000P_ooo`<0003Moooo0P0006[o ool300008_ooo`003oooo`800024oooo0P000=goool20000Joooo`80000R oooo000>oooo0`0008Coool30000foooo`80001/oooo0P0002;oool000ko ool20000Q_ooo`80003Koooo0P0006coool200008_ooo`003_ooo`800026 oooo0`000=Woool20000K_ooo`80000Qoooo000>oooo0P0008Ooool20000 fOooo`80001^oooo0P00027oool000goool20000R?ooo`<0003Goooo0`00 06koool200008Oooo`003Oooo`800029oooo0P000=Ooool20000Koooo`<0 000Poooo000=oooo0P0008Woool30000e_ooo`80001`oooo0P00023oool0 00coool20000Roooo`80003Eoooo0P00077oool200008?ooo`003?ooo`80 002;oooo0`000=Coool20000LOooo`80000Poooo000oooo0P000=7oool30000Looo o`80000Ooooo000;oooo0P0008ooool20000d?ooo`80001doooo0P0001oo ool000[oool30000Soooo`<0003>oooo0`0007Coool200007oooo`002_oo o`80002Aoooo0`000oooo0P0009Soool200006Ooo o`003?ooo`80002boooo0`0008coool30000V?ooo`80000Ioooo000Oooo`@0003Ooooo10000oooo`@0003Koooo100003oool20000:?ooo`00 I_ooo`H00025oooo1`000>;oool20000:Oooo`00JOooo`H0001ooooo1`00 0>Coool20000:_ooo`00K?ooo`L0001goooo1`000>Ooool30000:_ooo`00 K_ooo`T0001`oooo1`000>Woool30000:oooo`00L_ooo`P0001Yoooo1`00 0>coool30000;?ooo`00M_ooo`P0001Poooo2@000>koool30000;Oooo`00 N_ooo``0001Boooo30000?3oool30000;_ooo`00OOoooa400014oooo3000 0?Coool40000;oooo`00QOooo`d0000ioooo3@000?Woool30000"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{211.625, 196.75}, ImageMargins->{{74, 0}, {0, 0.625}}, ImageRegion->{{0, 1}, {0, 1}}]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Trefoil\ = \ X[a, d, b, e]\ X[e, b, f, c]\ X[c, f, d, a]\)], "Input"], Cell[BoxData[ \(X[a, d, b, e]\ X[c, f, d, a]\ X[e, b, f, c]\)], "Output"] }, Open ]], Cell[BoxData[{ RowBox[{ RowBox[{ "The", " ", "definition", " ", "of", " ", "the", " ", "Kauffman", " ", "Bracket", " ", "is", " ", "given", " ", "by", " ", "the", " ", "skein", " ", "relations", Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggoool20000aoooo`800035oooo0P000goool20000aoooo`800035oooo0P000_oool20000bOooo`800033oooo0P00 0_oool20000bOooo`80 0033oooo0P000Woool3 0000bOooo`<00031oooo0`000Woool20000boooo`800031oooo0P000<_oool20000:Oooo`00:_oo o`80003;oooo0P000>Woool20000boooo`800031oooo0P000<_oool20000 :Oooo`00:Oooo`80003=oooo0P000>Ooool20000cOooo`80002ooooo0P00 0Ooool20000cOooo`80 002ooooo0P000Goool3 0000cOooo`<0002moooo0`000Goool20000coooo`80002moooo0P000Goool20000coooo`80002moooo0P000?oool20000dOooo`80002koooo0P00 0=7oool200009_ooo`009oooo`80003Aoooo0P000>?oool20000dOooo`80 002koooo0P000=7oool200009_ooo`009_ooo`<0003Aoooo0`000>7oool3 0000dOooo`<0002ioooo0`000=7oool300009Oooo`009_ooo`80003Coooo 0P000>7oool20000doooo`80002ioooo0P000=?oool200009Oooo`009_oo o`80003Coooo0P000>7oool20000doooo`80002ioooo0P000=?oool20000 9Oooo`009Oooo`80003Eoooo0P000=ooool20000eOooo`80002goooo0P00 0=Goool200009?ooo`009Oooo`80003Eoooo0P000=ooool20000eOooo`80 002goooo0P000=Goool200009?ooo`009?ooo`<0000Joooo0P0009goool0 0`000?ooooooo`0Ioooo0`000=goool300006Oooo`@0002Koooo0`0001[o ool30000]Oooo`<0002ioooo0P0001[oool300008oooo`009?ooo`80000K oooo0`0001Goool40000H?ooo`@000000oooo`000000000Koooo0`0001_o ool20000gOooo`80000Joooo1@0001koool40000E_ooo`@000000oooo`00 0000000Joooo100001_oool20000]Oooo`80000Joooo0`0001Koool40000 J_ooo`@000000oooo`000000000Boooo0P0001_oool200008oooo`009?oo o`80000Loooo0`0001;oool70000GOooo`<00002oooo0`0001_oool30000 7?ooo`80003Moooo0P0001goool400006_ooo`L0001Coooo0`0000;oool3 00006Oooo`@0000Moooo0P000;Goool200006oooo`<0000Coooo1`0006Oo ool300000_ooo`<0000Boooo0`0001_oool200008oooo`008oooo`80000N oooo0`00017oool300001?ooo`030000oooooooo05[oool200001?ooo`80 000Joooo0`0001koool20000foooo`80000Ooooo1@0001Soool300001?oo o`030000oooooooo053oool200001?ooo`80000Goooo1000023oool20000 /oooo`80000Moooo0P0001?oool300001?ooo`030000oooooooo06Coool2 00001?ooo`80000Boooo0P0001goool200008_ooo`008oooo`80000Ooooo 0`0000ooool30000HOooo`800005oooo0P0001Woool300007oooo`80003K oooo0P00027oool500005Oooo`<0001Goooo0P0000Goool200005Oooo`@0 000Roooo0P000;?oool200007Oooo`<0000Aoooo0`0006_oool200001Ooo o`80000Aoooo0P0001koool200008_ooo`008_ooo`<0000Poooo0`0000ko ool20000H_ooo`800005oooo0P0001Soool300008?ooo`<0003Ioooo0`00 02?oool400005?ooo`80001Hoooo0P0000Goool200005?ooo`<0000Toooo 0`000;7oool300007_ooo`80000Aoooo0P0006coool200001Oooo`80000A oooo0P0001koool300008Oooo`008_ooo`80000Roooo0`0000goool20000 H_ooo`800005oooo0P0001Ooool300008_ooo`80003Ioooo0P0002Koool4 00004_ooo`80001Hoooo0P0000Goool200004_ooo`@0000Voooo0P000;7o ool200007oooo`<0000@oooo0P0006coool200001Oooo`80000@oooo0P00 023oool200008Oooo`008_ooo`80000Soooo0`0000coool20000H_ooo`80 0005oooo0P0001Koool300008oooo`80003Ioooo0P0002Ooool400004Ooo o`80001Hoooo0P0000Goool200004Oooo`<0000Xoooo0P000;7oool20000 8?ooo`80000@oooo0P0006coool200001Oooo`80000@oooo0P00023oool2 00008Oooo`008Oooo`80000Uoooo0`0000_oool20000H_ooo`800005oooo 0P0001Goool300009Oooo`80003Goooo0P0002[oool300004?ooo`80001H oooo0P0000Goool200004?ooo`<0000Zoooo0P000:ooool200008Oooo`<0 000?oooo0P0006coool200001Oooo`80000?oooo0P0002;oool200008?oo o`008Oooo`80000Voooo0`0000[oool20000H_ooo`800005oooo0P0001Co ool300009_ooo`80003Goooo0P0002_oool400003_ooo`80001Hoooo0P00 00Goool200003_ooo`@0000[oooo0P000:ooool200008_ooo`<0000>oooo 0P0006coool200001Oooo`80000>oooo0`0002;oool200008?ooo`008?oo o`<0000Woooo0`0000[oool00`000?ooooooo`03oooo0P0005coool20000 1?ooo`80000Coooo0`0002Ooool30000eOooo`<0000/oooo100000koool0 0`000?ooooooo`03oooo0P0005;oool200001?ooo`80000=oooo0`0002go ool30000[Oooo`<0000Soooo0P0000ooool00`000?ooooooo`03oooo0P00 06Koool200001?ooo`80000>oooo0P0002?oool300007oooo`008?ooo`80 000Yoooo0`0000Woool200000oooo`<0001Loooo0`0000;oool300004_oo o`<0000Yoooo0P000=Goool20000;oooo`<0000=oooo0P0000?oool30000 D_ooo`<00002oooo0`0000coool30000;oooo`80002]oooo0P0002Coool3 00003_ooo`800003oooo0`0006Koool300000_ooo`<0000=oooo0P0002Go ool200007oooo`008?ooo`80000Zoooo0`0000[oool50000Goooo`<00000 0oooo`000000000Aoooo0`0002[oool20000eOooo`80000`oooo100000go ool50000EOooo`<000000oooo`000000000:oooo1000033oool20000[Ooo o`80000Uoooo0`0000ooool50000JOooo`<000000oooo`000000000?ooo`<0001Uoooo0`00 00Goool30000>?ooo`80002Yoooo0P0002_oool30000O?ooo`<00008oooo 0`0002_oool200007Oooo`007Oooo`80000coooo0`0007Woool30000?ooo`80003=oooo0P0003ooool20000IOooo`80000ooooo0P000:Goool2 0000_ooo`80003;oooo0P00043oool30000Hoooo`800011 oooo0P000:?oool20000oooo`80003;oooo0P00047oool3 0000HOooo`<00011oooo0P000:?oool20000=?ooo`<0001koooo0`0003Co ool200006_ooo`006_ooo`<0000loooo0`0006_oool30000??ooo`<00039 oooo0`0004;oool30000Goooo`<00012oooo0`000:7oool30000=Oooo`<0 001ioooo0`0003Goool300006Oooo`006_ooo`80000noooo0`0006Woool3 0000?_ooo`800039oooo0P0004Coool20000Goooo`800014oooo0P000:7o ool20000=oooo`<0001goooo0`0003Ooool200006Oooo`006_ooo`80000o oooo0`0006Ooool30000?oooo`800039oooo0P0004Coool30000GOooo`80 0015oooo0P000:7oool20000>?ooo`<0001eoooo0`0003Soool200006Ooo o`006Oooo`800011oooo0`0006Goool30000@Oooo`800037oooo0P0004Ko ool30000Foooo`<00016oooo0P0009ooool20000>_ooo`<0001coooo0`00 03[oool200006?ooo`006Oooo`800012oooo0`0006?oool30000@_ooo`80 0037oooo0P0004Ooool30000FOooo`<00017oooo0P0009ooool20000>ooo o`@0001_oooo100003_oool200006?ooo`006?ooo`<00013oooo0`00067o ool30000@oooo`<00035oooo0`0004Soool20000FOooo`800018oooo0`00 09goool30000??ooo`@0001]oooo0`0003goool300005oooo`006?ooo`80 0015oooo0`0005ooool30000AOooo`800035oooo0P0004Woool30000Eooo o`80001:oooo0P0009goool20000?oooo`<0001[oooo0`0003ooool20000 5oooo`006?ooo`800016oooo0`0005goool30000A_ooo`800035oooo0P00 04[oool30000EOooo`<0001:oooo0P0009goool20000@?ooo`@0001Woooo 1000043oool200005oooo`005oooo`800018oooo0`0005_oool30000B?oo o`800033oooo0P0004coool20000EOooo`80001oooo0P0000Soool000Woool20000N_ooo`<0002@ oooo0P0004ooool50000Doooo`80001Voooo0P0003coool20000Ioooo`80 000Zoooo0P0005?oool20000ooooo`koool200002?ooo`002?ooo`<0001k oooo0`0008ooool30000D?ooo`D0001@oooo0`0006Koool30000>oooo`80 001Woooo0`0002Woool20000D_ooo`<0003ooooo3_ooo`<00007oooo0008 oooo0P0007goool30000Soooo`80001Boooo1@0001coool<00002_ooo``0 000@oooo0P0006Soool20000>oooo`80001Xoooo0P0002Woool200008ooo oaH0000Ioooo0P000?ooool@oooo0P0000Ooool000Soool20000O_ooo`<0 002>oooo0P0005Coool500006_ooo``0000:oooo3000013oool20000J?oo o`80000joooo0P0006Woool20000:Oooo`80000Soooo5P0001Woool20000 oooooa3oool200001oooo`001oooo`800020oooo0`0008koool20000EOoo o`D0000Loooo100001;oool400004oooo`80001Yoooo0P0003[oool20000 J_ooo`80000Xoooo0P0002Ooool400002_ooo`P0000Doooo0P000?oooolB oooo0P0000Koool000Ooool20000POooo`<0002=oooo0P0005Ooool50000 6_ooo`@0000Boooo100001?oool20000JOooo`80000joooo0P0006[oool2 0000:?ooo`80000Woooo100000[oool800005?ooo`80003ooooo4_ooo`80 0006oooo0006oooo0`0008;oool30000S?ooo`<0001Hoooo1@0001[oool2 00004_ooo`@0000Boooo0`0006Woool20000>_ooo`80001Zoooo0`0002Oo ool200009oooo`@0000>oooo100001?oool30000oooooa;oool300001Ooo o`001_ooo`800024oooo0`0008coool20000F_ooo`D0000Hoooo0P0001;o ool400004_ooo`80001Zoooo0P0003[oool20000Joooo`80000Woooo0P00 02Ooool400003_ooo`@0000Coooo0P000?oooolDoooo0P0000Goool000Ko ool20000QOooo`<0002;oooo0P0005coool500005_ooo`80000@oooo1000 01Coool20000J_ooo`80000joooo0P0006_oool200009oooo`80000Woooo 1000013oool400004Oooo`80003ooooo5?ooo`800005oooo0005oooo0P00 08Ooool30000Roooo`80001Moooo1@0001Coool200004?ooo`@0000Coooo 0P0006_oool20000>_ooo`80001/oooo0P0002Koool200009oooo`@0000@ oooo1000013oool20000oooooaKoool200001?ooo`001Oooo`800028oooo 0`0008[oool20000Goooo`D0000Boooo100000koool400004oooo`80001[ oooo0P0003[oool20000K?ooo`80000Voooo0P0002Ooool400004?ooo`@0 000@oooo0P000?oooolFoooo0P0000Coool000Coool30000ROooo`<00029 oooo0`0001;ooomC00004?ooo`@0000>oooo100001;oool30000Joooo`80 000joooo0P0006coool300004_ooobP0000Doooo1000013oool400003ooo o`<0003ooooo5_ooo`<00003oooo0005oooo0P0008[oool30000R?ooo`80 000CooooD@0001Coool200003_ooo`@0000Coooo0P0006_oool20000>_oo o`80001/oooo0P0001?ooolX00005?ooo`@0000@oooo1000013oool20000 oooooaKoool200001?ooo`001Oooo`80002;oooo0`0008Ooool20000GOoo o`D0000Foooo0P0000koool400004oooo`80001[oooo0P0003[oool20000 K?ooo`80000Voooo0P0002Ooool400004?ooo`@0000@oooo0P000?oooolF oooo0P0000Coool000Koool20000Roooo`<00025oooo0P0005coool50000 6?oooa80000Foooo0P0006[oool20000>_ooo`80001[oooo0P0002Ooool2 00009oooo`@0000>oooo100001?oool20000oooooaCoool200001Oooo`00 1_ooo`80002_ooo`80001Zoooo0`0002Ooool200009oooo`@0000oooo0`0007koool20000E?ooo`D0000Toooo 0P0000[oool400006?ooo`80001Xoooo0P0003[oool20000JOooo`80000Y oooo0P0002OooolB00006Oooo`80003ooooo4?ooo`800007oooo0008oooo 0P0008ooool30000OOooo`80001Boooo1@0002Koool200002?ooo`@0000J oooo0P0006Soool20000>_ooo`<0001Xoooo0P0002Woool200009oooo`@0 000:oooo0P0001_oool20000oooooa3oool200001oooo`002?ooo`<0002? oooo0`0007_oool30000D?ooo`D0000Xoooo0P0000Soool400006_ooo`<0 001Woooo0P0003_oool20000Ioooo`<0000Yoooo0P0002Ooool400002_oo o`80000Koooo0`000?ooool>oooo0`0000Ooool000Woool20000T?ooo`<0 001joooo0P0004ooool50000:_ooo`@00006oooo100001_oool20000I_oo o`80000loooo0P0006Ooool20000:_ooo`80000Woooo100000coool40000 6?ooo`80003ooooo3_ooo`800008oooo0009oooo0P00097oool30000NOoo o`80001=oooo1@0002coool400001_ooo`@0000Koooo0P0006Koool20000 ??ooo`80001Woooo0P0002[oool200009oooo`@0000oooo100001Ooool30000ooooo`[oool30000 2Oooo`002oooo`80002Coooo0`0007?oool20000Aoooo`D0000foooo0P00 00Coool400007oooo`80001Toooo0P0003coool30000I?ooo`80000/oooo 0P0002Ooool400003_ooo`@0000Hoooo0P000?ooool:oooo0P0000[oool0 00_oool20000U?ooo`<0001boooo0P0004Goool50000>_ooo`800002oooo 100001ooool20000I?ooo`80000moooo0P0006Coool20000;?ooo`80000W oooo100000koool400006?ooo`80003ooooo2_ooo`80000:oooo000oooo100001Woool20000ooooo`Soool200002oooo`003?ooo`80002E oooo0`0006ooool20000@_ooo`D0000noooo1P0002;oool20000H_ooo`80 000noooo0P0006?oool20000;Oooo`80000Woooo100000coool400006ooo o`80003ooooo2?ooo`80000;oooo000oooo0P0006Woool30000 :oooo`<0001Yoooo0P0008Woool200009_ooo`80001Ooooo0P00043oool2 0000H?ooo`80001Doooo5000027oool20000ooooo`Coool200003Oooo`00 3_ooo`80001Xoooo0`0002goool30000J?ooo`800029oooo0P0002Koool2 0000Goooo`800010oooo0`0005ooool20000E?oooa@0000Qoooo0P000?oo ool4oooo0P0000goool000koool30000I_ooo`<0000_oooo0`0006Koool3 0000/Oooo`<0001Noooo0P00047oool20000G_ooo`<00029oooo0`000?oo ool2oooo0`0000goool000ooool20000IOooo`<0000aoooo0`0006Goool2 0000/oooo`80001Moooo0P0004;oool20000G_ooo`80002;oooo0P000?oo ool2oooo0P0000koool000ooool20000I?ooo`<0000coooo0`0006Coool2 0000/oooo`80001Moooo0P0004;oool30000GOooo`80002;oooo0P000?oo ool2oooo0P0000koool0013oool20000H_ooo`<0000eoooo0`0006;oool2 0000]Oooo`80001Loooo0P0004?oool20000G?ooo`80002=oooo0P000?oo ool200003oooo`004?ooo`80001Qoooo0`0003Ooool30000HOooo`80002e oooo0P0005_oool20000A?ooo`80001Loooo0P0008goool20000ooooo`80 000?oooo000@oooo0`0005ooool30000>Oooo`<0001Ooooo0`000;Goool3 0000F_ooo`800014oooo0`0005[oool30000SOooo`<0001doooo5@0007Co ool300003oooo`004Oooo`80001Noooo0`0003_oool30000G_ooo`80002g oooo0P0005[oool20000AOooo`80001Joooo0P0008ooool20000KOooob<0 001]oooo0P00013oool0017oool20000GOooo`<0000moooo0`0005goool2 0000]oooo`80001Ioooo0P0004Koool20000F_ooo`80002?oooo0P0006Wo ool;00005?ooo``0001Yoooo0P00013oool001;oool20000Foooo`<0000o oooo0`0005_oool20000^Oooo`80001Hoooo0P0004Koool30000F?ooo`80 002Aoooo0P0006Coool800008_ooo`T0001Toooo0P00017oool001;oool2 0000F_ooo`<00011oooo0`0005[oool20000^Oooo`80001Hoooo0P0004Oo ool20000F?ooo`80002Aoooo0P00067oool70000:_ooo`P0001Qoooo0P00 017oool001;oool30000F?ooo`<00013oooo0`0005Soool30000^Oooo`<0 001Foooo0P0004Soool30000E_ooo`<0002Aoooo0`0005goool60000<_oo o`L0001Moooo0`00017oool001?oool20000Eoooo`<00015oooo0`0005Oo ool20000^oooo`80001Foooo0P0004Woool20000E_ooo`80002Coooo0P00 05[oool60000>?ooo`L0001Joooo0P0001;oool001?oool20000E_ooo`<0 0017oooo0`0005Koool20000^oooo`80001Eoooo0P0004[oool30000EOoo o`80002Coooo0P0005Soool50000?_ooo`H0001Hoooo0P0001;oool001Co ool20000E?ooo`<00019oooo0`0005Coool20000_Oooo`80001Doooo0P00 04_oool20000E?ooo`80002Eoooo0P0005Goool40000A?ooo`D0001Eoooo 0P0001?oool001Coool20000Doooo`<0001;oooo0`0005?oool20000_Ooo o`80001Coooo0P0004coool20000E?ooo`80002Eoooo0P0005;oool50000 B?ooo`H0001Boooo0P0001?oool001Coool30000DOooo`<0001=oooo0`00 057oool30000_Oooo`<0001Boooo0P0004coool30000D_ooo`<0002Eoooo 0`00053oool40000C?ooo`D0001@oooo0`0001?oool001Goool20000D?oo o`<0001?oooo0`00053oool20000_oooo`80001Aoooo0`0004goool30000 DOooo`80002Goooo0P0004koool40000DOooo`@0001>oooo0P0001Coool0 01Goool20000Coooo`<0001Aoooo0`0004ooool20000_oooo`80001Aoooo 0P0004ooool20000DOooo`80002Goooo0P0004coool40000E?ooo`D0001< oooo0P0001Coool001Koool20000COooo`<0001Coooo0`0004goool20000 `Oooo`80001?oooo0P00053oool30000Coooo`80002Ioooo0P0004Woool4 0000F?ooo`D00019oooo0P0001Goool001Koool20000C?ooo`<0001Eoooo 0`0004coool20000`Oooo`80001?oooo0P00057oool20000Coooo`80002I oooo0P0004Soool30000G?ooo`@00018oooo0P0001Goool001Koool30000 B_ooo`<0001Goooo0`0004[oool30000`Oooo`<0001=oooo0P0005;oool3 0000COooo`<0002Ioooo0`0004Goool40000Goooo`@00015oooo0`0001Go ool001Ooool20000BOooo`<0001Ioooo0`0004Woool20000`oooo`80001= oooo0P0005?oool20000COooo`80002Koooo0P0004Coool30000H_ooo`@0 0014oooo0P0001Koool001Ooool20000B?ooo`<0001Koooo0`0004Soool2 0000`oooo`80001ooo o`<0001]oooo0`0003_oool20000boooo`800011oooo0`00067oool30000 @Oooo`80002Soooo0P0003Coool30000Noooo`<0000doooo0P0001[oool0 01_oool20000>_ooo`<0001_oooo0`0003[oool20000boooo`800011oooo 0P0006?oool20000@Oooo`80002Soooo0P0003?oool30000OOooo`<0000c oooo0P0001[oool001coool20000>?ooo`<0001aoooo0`0003Soool20000 cOooo`80000ooooo0P0006Coool30000?oooo`80002Uoooo0P00037oool3 0000Ooooo`<0000aoooo0P0001_oool001coool20000=oooo`<0001coooo 0`0003Ooool20000cOooo`80000noooo0`0006Goool30000?_ooo`80002U oooo0P00033oool30000POooo`<0000`oooo0P0001_oool001coool30000 =Oooo`<0001eoooo0`0003Goool30000cOooo`<0000loooo0`0006Ooool3 0000??ooo`<0002Uoooo0`0002koool30000Poooo`<0000^oooo0`0001_o ool001goool20000=?ooo`<0001goooo0`0003Coool20000coooo`80000k oooo0`0006Woool30000>oooo`80002Woooo0P0002koool20000QOooo`80 000^oooo0P0001coool001goool20000_ooo`80002Woooo0P0002go ool20000Q_ooo`<0000]oooo0P0001coool001koool20000?ooo`80 002Yoooo0P0002_oool30000Qoooo`<0000[oooo0P0001goool001koool2 0000oooo0P00 00Goool20000F_ooo`800004oooo0P0001Soool300008_ooo`80003Ioooo 0P0002Koool400004oooo`800005oooo0P00053oool200001?ooo`80000C oooo100002Koool20000/Oooo`80000Poooo0P00017oool200001Oooo`80 001Toooo0P0000Coool200004Oooo`80000Poooo0P00027oool002;oool3 00008?ooo`<0000?oooo0P0000Goool20000F_ooo`800004oooo0P0001Wo ool300008?ooo`<0003Ioooo0`0002Coool300005Oooo`800005oooo0P00 053oool200001?ooo`80000Doooo100002Coool30000/Oooo`<0000Noooo 0P0001;oool200001Oooo`80001Toooo0P0000Coool200004Oooo`<0000N oooo0`00027oool002?oool200007oooo`<0000@oooo0P0000Goool20000 Foooo`800003oooo0P0001[oool300007oooo`80003Koooo0P0002;oool4 00005_ooo`800005oooo0P00057oool200000oooo`80000Foooo100002;o ool20000/oooo`80000Noooo0P0001;oool200001Oooo`80001Uoooo0P00 00?oool200004_ooo`80000Noooo0P0002;oool002?oool200007_ooo`<0 000Aoooo0P0000Goool20000G?ooo`<000000oooo`000000000Koooo0`00 01koool20000foooo`80000Poooo100001Soool200001Oooo`80001Boooo 0`000003oooo0000000001Ooool500008?ooo`80002coooo0P0001goool2 00004oooo`800005oooo0P0006Koool3000000?oool0000000004_ooo`<0 000Moooo0P0002;oool002Coool200007?ooo`<0000Boooo0P0000Goool2 0000G_ooo`@0000Loooo0`0001coool20000gOooo`80000Moooo100001[o ool200001Oooo`80001Doooo100001Woool500007Oooo`80002eoooo0P00 01_oool300004oooo`800005oooo0P0006Soool400004oooo`<0000Koooo 0P0002?oool002Coool200006oooo`<0000Coooo0P0000Goool20000H?oo o`80000Moooo0`0001_oool20000gOooo`80000Loooo0`0001coool20000 1Oooo`80001Foooo0P0001_oool400007?ooo`80002eoooo0P0001_oool2 00005?ooo`800005oooo0P0006[oool200005?ooo`80000Koooo0P0002?o ool002Coool300006_ooo`80000Doooo0P0000Coool20000G?ooo`800003 oooo0P0001koool00`000?ooooooo`0Ioooo0`000=goool300006_ooo`<0 000Moooo0P0000Coool20000D_ooo`800003oooo0P0001goool200006ooo o`<0002eoooo0`0001Woool200005Oooo`800004oooo0P0006Koool20000 0oooo`80000Doooo0P0001[oool300008oooo`009Oooo`80000`oooo0`00 00;oool30000G?ooo`800002oooo0`0003[oool20000goooo`80000Joooo 00<0003oooooool07Oooo`<00002oooo0`0005;oool200000_ooo`<0000N oooo00<0003oooooool06Oooo`80002goooo0P00033oool300000_ooo`<0 001Voooo0P0000;oool300005?ooo`80000Joooo0P0002Coool002Goool2 0000oooo`80003Ooooo 0P0003[oool2000000Coool000000000001Eoooo1@0003_oool20000]ooo o`80000`oooo0P000004oooo000000000000JOooo`D0000aoooo0P0002Co ool002Koool20000;oooo`80002Roooo0P000>7oool20000>Oooo`80002H oooo0P000;Woool20000;oooo`80002Roooo0P0002Goool002Koool20000 ;oooo`80002Roooo0P000>7oool20000>Oooo`80002Hoooo0P000;Woool2 0000;oooo`80002Roooo0P0002Goool002Koool30000;_ooo`80002Qoooo 0`000>7oool30000>?ooo`80002Goooo0`000;Woool30000;_ooo`80002Q oooo0`0002Goool002Ooool20000;_ooo`80002Qoooo0P000>?oool20000 >?ooo`80002Goooo0P000;_oool20000;_ooo`80002Qoooo0P0002Koool0 02Ooool20000;Oooo`<0002Qoooo0P000>?oool20000=oooo`<0002Goooo 0P000;_oool20000;Oooo`<0002Qoooo0P0002Koool002Soool20000cooo o`80003Uoooo0P000goool00`000?oooooo o`36oooo0P000"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{490, 96.5625}, ImageMargins->{{0, 0}, {0, 5}}, ImageRegion->{{0, 1}, {0, 1}}]}], " "}], "\[IndentingNewLine]", "and", "\[IndentingNewLine]", Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooo`<0000Woooo0`0001goool30000]Oooo`<0000[oooo0`0001Go ool001Ooool20000>_ooo`@0000Yoooo100001coool20000]Oooo`80000] oooo0P0001Goool001Koool30000>Oooo`<0000]oooo0`0001_oool30000 /oooo`<0000]oooo0`0001Coool001Koool20000>Oooo`<0000_oooo0`00 01_oool20000/oooo`80000_oooo0P0001Coool001Goool30000>?ooo`<0 000aoooo0`0001[oool30000/Oooo`<0000_oooo0`0001?oool001Goool2 0000>?ooo`80000eoooo0P0001[oool20000/Oooo`80000aoooo0P0001?o ool001Coool30000=oooo`80000goooo0P0001Woool30000[oooo`<0000a oooo0`0001;oool001Coool20000=oooo`80000ioooo0P0001Woool20000 [oooo`80000coooo0P0001;oool001?oool30000=_ooo`80000koooo0P00 01Soool30000[Oooo`<0000coooo0`00017oool001?oool20000=_ooo`<0 000koooo0`0001Soool20000[Oooo`80000eoooo0P00017oool001;oool3 0000=Oooo`<0000moooo0`0001Ooool30000Zoooo`<0000eoooo0`00013o ool001;oool20000=_ooo`80000ooooo0P0001Soool20000Zoooo`80000g oooo0P00013oool0017oool30000=Oooo`800011oooo0P0001Ooool30000 ZOooo`<0000goooo0`0000ooool0017oool20000=Oooo`<00011oooo0`00 01Ooool20000ZOooo`80000ioooo0P0000ooool0013oool30000=Oooo`80 0013oooo0P0001Ooool30000A?ooo`<0001Poooo0`0003Woool300003_oo o`004?ooo`80000@oooo300001Woool20000AOooo`80000Goooo0P0004Co ool50000?Oooo`P00002oooo100001?oool200004?ooo``0000Ooooo0P00 00koool000ooool300004?ooo``0000Ioooo0P0004Goool200005oooo`<0 0015oooo1@0003_oool800000_ooo`@0000Boooo0`00013oool<00007ooo o`<0000=oooo000?oooo0P0001Goool400007?ooo`800017oooo0P0001Oo ool20000Aoooo`D0000eoooo1P0000Coool600005?ooo`80000Eoooo1000 02Coool200003Oooo`003_ooo`<0000Eoooo100001coool20000Aoooo`80 000Goooo0`0004Soool50000oooo1@0002Woool400002_oo o`@0000Aoooo0`0001Ooool400009_ooo`<0000:oooo000oooo0`0001[oool40000 :Oooo`<00007oooo0009oooo0P0001_oool400006Oooo`80001=oooo0P00 01[oool20000FOooo`D0000Koooo100000[oool400003_ooo`80000Koooo 100002[oool200001oooo`002?ooo`<0000Koooo100001Woool20000COoo o`80000Joooo0`0005[oool500006Oooo`@0000:oooo100000goool30000 6oooo`@0000Zoooo0`0000Koool000Soool200007?oooa00000=oooo0P00 04goool200006oooo`80001Loooo1@0001Ooool400002_ooo`@0000=oooo 0P0001coool@00007oooo`800006oooo0007oooo0`0001coool@00003Ooo o`80001=oooo0P0001_oool30000GOooo`D0000Eoooo100000[oool40000 3?ooo`<0000Loooo400001ooool300001Oooo`001oooo`80000Moooo1000 00[oool600002Oooo`80001=oooo0P0001coool20000Goooo`D0000Eoooo 100000Soool400003?ooo`80000Moooo100000[oool600007?ooo`800005 oooo0006oooo0`0001goool400002_ooo`H00008oooo0P0004koool20000 7?ooo`<0000BooooD`0001?oool400002?ooo`@0000;oooo0`0001goool4 00002_ooo`H0000Loooo0`0000Coool000Ooool200007Oooo`@0000oooo100000Ooool20000COooo`80000Joooo0`0005[o ool500007oooo`H00002oooo100000goool300006oooo`@0000>oooo1000 01Soool300001_ooo`002Oooo`80000Koooo100000koool400001oooo`80 001=oooo0P0001[oool20000FOooo`D0000Yoooo100000koool200006ooo o`@0000>oooo100001Soool200001oooo`002Oooo`<0000Joooo100000ko ool400001oooo`80001=oooo0P0001Woool30000Eoooo`D0000[oooo1000 00koool300006_ooo`@0000>oooo100001Ooool300001oooo`002_ooo`80 000Joooo100000koool400001oooo`80001=oooo0P0001Woool20000E_oo o`D0000]oooo100000ooool200006_ooo`@0000>oooo100001Ooool20000 2?ooo`002_ooo`<0000Ioooo100000koool400001oooo`80001=oooo0P00 01Soool30000E?ooo`D0000_oooo100000ooool300006Oooo`@0000>oooo 100001Koool300002?ooo`002oooo`80000Ioooo100000coool400002Ooo o`<0001;oooo0P0001Woool20000Doooo`D0000aoooo1000013oool20000 6Oooo`@0000oooo`80 000>oooo000@oooo0`0003Goool20000@oooo`80000Goooo0`0004Coool3 0000H?ooo`<0000ioooo0`0000koool0017oool20000=Oooo`<00011oooo 0`0001Ooool20000ZOooo`80000ioooo0P0000ooool0017oool30000=Ooo o`800011oooo0P0001Ooool30000ZOooo`<0000goooo0`0000ooool001;o ool20000=_ooo`80000ooooo0P0001Soool20000Zoooo`80000goooo0P00 013oool001;oool30000=Oooo`<0000moooo0`0001Ooool30000Zoooo`<0 000eoooo0`00013oool001?oool20000=_ooo`<0000koooo0`0001Soool2 0000[Oooo`80000eoooo0P00017oool001?oool30000=_ooo`80000koooo 0P0001Soool30000[Oooo`<0000coooo0`00017oool001Coool20000=ooo o`80000ioooo0P0001Woool20000[oooo`80000coooo0P0001;oool001Co ool30000=oooo`80000goooo0P0001Woool30000[oooo`<0000aoooo0`00 01;oool001Goool20000>?ooo`80000eoooo0P0001[oool20000/Oooo`80 000aoooo0P0001?oool001Goool30000>?ooo`<0000aoooo0`0001[oool3 0000/Oooo`<0000_oooo0`0001?oool001Koool20000>Oooo`<0000_oooo 0`0001_oool20000/oooo`80000_oooo0P0001Coool001Koool30000>Ooo o`<0000]oooo0`0001_oool30000/oooo`<0000]oooo0`0001Coool001Oo ool20000>_ooo`@0000Yoooo100001coool20000]Oooo`80000]oooo0P00 01Goool001Ooool30000>oooo`<0000Woooo0`0001goool30000]Oooo`<0 000[oooo0`0001Goool001Soool20000??ooo`@0000Soooo100001koool2 0000]oooo`80000[oooo0P0001Koool001Soool30000?Oooo`@0000Noooo 1@0001ooool30000]oooo`<0000Yoooo0`0001Koool001Woool20000?ooo o`D0000Ioooo1@00027oool20000^Oooo`80000Yoooo0P0001Ooool001Wo ool30000@?ooo`L0000Aoooo1`0002;oool30000^Oooo`<0000Woooo0`00 01Ooool001[oool20000@oooo``000000oooo`000000000:00009Oooo`80 002koooo0P0002Ooool200006?ooo`006_ooo`030000oooooooo04GooolB 0000:Oooo`80002koooo00<0003oooooool09_ooo`80000Hoooo001[oooo 00<0003oooooool0oooooc3oool00?oooonNoooo003oooooW_ooo`00oooo oikoool00?oooonNoooo003oooooW_ooo`00oooooikoool00?oooonNoooo 0000\ \>"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{191.625, 44.125}, ImageMargins->{{111.625, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], "\[IndentingNewLine]", \(The\ first\ rule\ can\ be\ coded\ as\ \(\ \(follows\)\(:\)\)\)}], "Text", Evaluatable->False, FontFamily->"Times New Roman"], Cell[BoxData[ \(\(rule1\ = \ \((X[a_, b_, c_, d_]\ :> \ A\ \[Delta][a\ d] \[Delta][b\ c]\ + \ B\ \[Delta][a\ b] \[Delta][c\ d])\);\)\)], "Input"], Cell["\<\ Let as apply the first rule to the trefoil knot, just as an example:\ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(Trefoil\ /. \ rule1\)], "Input"], Cell[BoxData[ \(\((A\ \[Delta][b\ d]\ \[Delta][a\ e] + B\ \[Delta][a\ d]\ \[Delta][b\ e])\)\ \((A\ \[Delta][ c\ e]\ \[Delta][b\ f] + B\ \[Delta][b\ e]\ \[Delta][c\ f])\)\ \((B\ \[Delta][ a\ d]\ \[Delta][c\ f] + A\ \[Delta][a\ c]\ \[Delta][d\ f])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Trefoil\ /. \ rule1\ // \ Expand\)], "Input"], Cell[BoxData[ \(A\^2\ B\ \[Delta][a\ d]\ \[Delta][b\ d]\ \[Delta][a\ e]\ \[Delta][ c\ e]\ \[Delta][b\ f]\ \[Delta][c\ f] + A\ B\^2\ \[Delta][a\ d]\^2\ \[Delta][b\ e]\ \[Delta][c\ e]\ \[Delta][ b\ f]\ \[Delta][c\ f] + A\ B\^2\ \[Delta][a\ d]\ \[Delta][b\ d]\ \[Delta][a\ e]\ \[Delta][ b\ e]\ \[Delta][c\ f]\^2 + B\^3\ \[Delta][a\ d]\^2\ \[Delta][b\ e]\^2\ \[Delta][c\ f]\^2 + A\^3\ \[Delta][a\ c]\ \[Delta][b\ d]\ \[Delta][a\ e]\ \[Delta][ c\ e]\ \[Delta][b\ f]\ \[Delta][d\ f] + A\^2\ B\ \[Delta][a\ c]\ \[Delta][a\ d]\ \[Delta][b\ e]\ \[Delta][ c\ e]\ \[Delta][b\ f]\ \[Delta][d\ f] + A\^2\ B\ \[Delta][a\ c]\ \[Delta][b\ d]\ \[Delta][a\ e]\ \[Delta][ b\ e]\ \[Delta][c\ f]\ \[Delta][d\ f] + A\ B\^2\ \[Delta][a\ c]\ \[Delta][a\ d]\ \[Delta][b\ e]\^2\ \[Delta][ c\ f]\ \[Delta][d\ f]\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ "We", " ", "see", " ", "that", " ", "we", " ", "need", " ", "a", " ", "rule", " ", "that", " ", "will", " ", "dispose", " ", "of", " ", "unnecessary", " ", "\<\"intermidiate\"\>", " ", RowBox[{ "points", ":", "\[IndentingNewLine]", Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgQP00`40o`00 0?l0003ooooooooooooooooo0oooo`00oooooooooooooooo0oooo`00oooo oooooooooooo0oooo`00oooooooooooooooo0oooo`00oooooooooooooooo 0oooo`00oooooooooooooooo0oooo`00oooooooooooooooo0oooo`00oooo oooooooooooo0oooo`00AOoooad0003oooooF_ooo`<0003ooooo@oooo`00 ??ooobl0001/oooo00<0003oool000000`0007koool40000G?ooo`D0003o oooo@Oooo`00=Ooooa00000Loooo4@0006Goool300000_ooo`<0001joooo 1`0005goool50000ooooocooool0033oool<0000;_ooo`d0001Poooo0P00 00Coool20000N_ooo`<00004oooo00<0003oooooool0G?ooo`D0003ooooo ?Oooo`00;?ooo`T0000loooo2P0005coool200001Oooo`80001hoooo0`00 06Koool50000oooooc_oool002Soool80000A_ooo`T0001Hoooo0P0000Go ool20000N?ooo`80001Yoooo1@000?oooolioooo000Toooo200004koool9 0000E?ooo`800005oooo0P0007Soool20000Joooo`D0003ooooo=oooo`00 8Oooo`L0001Foooo2000057oool200001Oooo`80001hoooo0P0006goool5 0000ooooocGoool001goool70000G_ooo`P0001=oooo0P0000Goool20000 N?ooo`80001_oooo1@000?oooolcoooo000Joooo1`0006Coool80000B_oo o`800005oooo0P0007Soool20000LOooo`D0003ooooo?ooo`H00027oooo1@0007?o ool50000ooooobGoool000ooool200000oooo`80001koooo0P0000Goool2 0000>oooo`D00023oooo100007Coool50000ooooobOoool0013oool30000 00?oool000000000Noooo`800005oooo0P0003goool60000OOooo`D0001d oooo1@000?oooolYoooo000Boooo100007_oool200001Oooo`80000ooooo 1P0007Woool50000M?ooo`D0003ooooo:oooo`005?ooo`80001koooo0P00 00Goool20000@_ooo`H0001coooo1@0007Goool50000ooooobgoool000oo ool200000oooo`80001koooo0P0000Coool20000AOooo`H0001_oooo1@00 07Goool50000ooooobooool000ooool200000_ooo`<0001koooo0`0000;o ool30000B?ooo`H0001Yoooo1@0007Koool50000?Oooo`<00003oooo0P00 0>3oool400001oooo`004?ooo`D0001loooo0P000004oooo000000000000 C?ooo`P0001Qoooo1`0007Koool50000?_ooo`D000000oooo`000000003O oooo1`0000Koool0097oool20000Doooo`P0001Koooo1`0007Ooool50000 @?ooo`800003oooo0`000=ooool300001?ooo`030000oooooooo00?oool0 097oool20000Eoooo`P0001Coooo1`0007Woool50000@_ooo`800004oooo 0P000=koool300002oooo`00TOooo`80001Joooo2@0004_oool80000N_oo o`D00014oooo0P0000Coool20000g_ooo`80000"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{490, 37}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}]}]}]], "Text", Evaluatable->False, FontFamily->"Times New Roman"], Cell[BoxData[ \(\(rule2\ = \ \((\[Delta][a_\ b_] \[Delta][b_\ c_]\ :> \ \[Delta][ a\ c])\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\((Trefoil\ /. \ rule1\ // \ Expand)\)\ //. \ rule2\)], "Input"], Cell[BoxData[ \(A\^2\ B\ \[Delta][c\ f]\^2 + A\ B\^2\ \[Delta][a\ d]\^2\ \[Delta][c\ f]\^2 + A\ B\^2\ \[Delta][b\ e]\^2\ \[Delta][c\ f]\^2 + B\^3\ \[Delta][a\ d]\^2\ \[Delta][b\ e]\^2\ \[Delta][c\ f]\^2 + 2\ A\^2\ B\ \[Delta][d\ f]\^2 + A\ B\^2\ \[Delta][b\ e]\^2\ \[Delta][d\ f]\^2 + A\^3\ \[Delta][c\ e]\^2\ \[Delta][d\ f]\^2\)], "Output"] }, Open ]], Cell["\<\ This done, we can code the second rule, (using dd instead of d, to avoid \ naming conflicts)\ \>", "Text", Evaluatable->False], Cell[BoxData[ \(\(rule3\ = \ {\((\[Delta][_])\)^2\ -> \ dd, \ \[Delta][_^2]\ \[Rule] \ dd};\)\)], "Input"], Cell["and see what heppens to the trefoil knot:", "Text", Evaluatable->False], Cell[BoxData[ \(RawBracket[t_]\ := \ Simplify[\(\((t /. rule1\ // \ Expand)\)\ //. \ rule2\)\ /. \ rule3]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(RawBracket[Trefoil]\)], "Input"], Cell[BoxData[ \(dd\ \((3\ A\^2\ B + A\^3\ dd + 3\ A\ B\^2\ dd + B\^3\ dd\^2)\)\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{\(Let'\), "s", " ", "verify", " ", "the", " ", "second", " ", "Reidemeister", " ", RowBox[{ "move", ":", "\[IndentingNewLine]", Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooo`/0000Eoooo0`0000;oool300007?ooo`80002eoooo0P00 01goool400006_ooo`L0001Coooo0`0000;oool300006Oooo`@0000Moooo 0P0002[oool002goool200007_ooo`<00004oooo00<0003oooooool06?oo o`T0000coooo200001coool200001?ooo`80000Moooo0P000;?oool20000 7oooo`D0000Hoooo0`0000Coool00`000?ooooooo`1@oooo0P0000Coool2 00005oooo`@0000Poooo0P0002Woool002goool200007Oooo`<0000Toooo 200002goool700007oooo`800005oooo0P0001goool20000/oooo`80000Q oooo1@0001Goool30000Eoooo`800005oooo0P0001Goool400008_ooo`80 000Yoooo000/oooo0`0001goool20000:Oooo`L0000Woooo1P0002?oool2 00001Oooo`80000Moooo0`000;7oool300008oooo`@0000Doooo0P0005So ool200001Oooo`80000Doooo0`0002Coool30000:?ooo`00;?ooo`80000N oooo0P0002coool600008oooo`D0000Voooo0P0000Goool200007_ooo`80 002aoooo0P0002Koool400004_ooo`80001Hoooo0P0000Goool200004_oo o`@0000Voooo0P0002Soool002coool200007_ooo`80000_oooo1@0001oo ool40000:Oooo`800005oooo0P0001koool20000/Oooo`80000Woooo1000 017oool20000F?ooo`800005oooo0P00017oool30000:?ooo`80000Xoooo 000[oooo0P0001ooool20000oooo00<0003oooooool00oooo`80001Boooo0P0000Co ool200003Oooo`<0000]oooo0`0002Koool002[oool200008Oooo`800003 oooo0`00033oool500003oooo`@0000boooo0`0000;oool300008?ooo`80 002]oooo0P0002ooool300003Oooo`800003oooo0`0005;oool300000_oo o`<0000_ooo`<0001[oooo0`0003[oool200008oooo`009oooo`80001X oooo100006Goool20000Yoooo`80000koooo0`0006Woool30000>oooo`80 000Soooo000Voooo0`0006[oool30000I?ooo`<0002Uoooo0`0003coool3 0000Ioooo`<0000loooo0`0002;oool002Koool20000K?ooo`<0001Toooo 0P000:Goool20000?_ooo`<0001Uoooo0`0003koool200008_ooo`009_oo o`80001]oooo0`0006?oool20000YOooo`80000ooooo0P0006Goool20000 ?oooo`80000Roooo000Uoooo0P0006ooool30000Hoooo`80002Soooo0P00 043oool30000Hoooo`800011oooo0P00027oool002Goool20000L?ooo`<0 001Roooo0P000:?oool20000@Oooo`<0001Qoooo0`00047oool200008Ooo o`009?ooo`<0001Qoooo00<0003oooooool03Oooo`<0001Qoooo0`000:7o ool30000@_ooo`<0001Ooooo0`0004;oool300008?ooo`009?ooo`80001Q oooo0P00013oool30000HOooo`80002Qoooo0P0004Coool20000Goooo`80 0014oooo0P00023oool002Coool20000H?ooo`<0000Aoooo0`00063oool2 0000XOooo`800014oooo0`0005goool20000AOooo`80000Poooo000Soooo 0P00063oool300004oooo`<0001Poooo0P0009ooool20000A_ooo`<0001K oooo0`0004Koool200007oooo`008oooo`80001Poooo0P0001Goool20000 H?ooo`80002Ooooo0P0004Ooool30000FOooo`<00017oooo0P0001ooool0 02;oool30000Goooo`80000Foooo0`0005ooool30000WOooo`<00018oooo 0P0005Woool20000B?ooo`<0000Noooo000Roooo0P0005ooool300005ooo o`<0001Ooooo0P0009goool20000BOooo`<0001Goooo0P0004[oool20000 7_ooo`008_ooo`80001Noooo0`0001Woool30000G_ooo`80002Moooo0P00 04[oool30000EOooo`<0001:oooo0P0001koool0027oool20000Goooo`80 000Koooo0P0005ooool20000Voooo`80001oooo0`00057oool20000Coooo`80000Loooo000Poooo 0P0005goool200008?ooo`<0001Moooo0P0009Woool20000Coooo`80001A oooo0P0004ooool200007?ooo`007oooo`80001Moooo0`00027oool30000 GOooo`80002Goooo0P00053oool30000Coooo`80001Aoooo0P0001_oool0 01ooool20000GOooo`80000Soooo0P0005goool20000Uoooo`80001Aoooo 0`0004goool30000DOooo`80000Koooo000Noooo0`0005coool200009?oo o`<0001Loooo0`0009Goool30000D_ooo`80001=oooo0P0005;oool30000 6_ooo`007_ooo`80001Moooo0P0002Goool20000GOooo`80002Eoooo0P00 05?oool20000C?ooo`80001Doooo0P0001[oool001koool20000G?ooo`80 000Voooo0`0005coool20000UOooo`80001Coooo0`0004_oool20000E?oo o`80000Joooo000Moooo0P0005goool200009oooo`80001Moooo0P0009?o ool20000EOooo`80001;oooo0P0005Goool200006Oooo`007Oooo`80001L oooo0P0002Soool30000G?ooo`80002Coooo0P0005Goool30000BOooo`80 001Foooo0P0001Woool001coool30000G?ooo`80000Yoooo0P0005coool3 0000TOooo`<0001Foooo0P0004Woool20000E_ooo`<0000Hoooo000Loooo 0P0005coool20000:_ooo`<0001Loooo0P00097oool20000Eoooo`<00017 oooo0P0005Soool200006?ooo`007?ooo`80001Loooo0P0002_oool20000 G?ooo`80002Aoooo0P0005Soool20000Aoooo`80001Hoooo0P0001Soool0 01_oool20000G?ooo`80000/oooo0`0005coool20000Soooo`80001Ioooo 0P0004Koool20000F_ooo`80000Goooo000Koooo0P0005coool20000;Ooo o`80001Loooo0P0008ooool20000FOooo`<00015oooo0P0005[oool20000 5oooo`006_ooo`<0001Koooo0P0002koool30000Foooo`<0002=oooo0`00 05[oool20000AOooo`80001Joooo0`0001Koool001[oool20000G?ooo`80 000_oooo0P0005coool20000SOooo`80001Koooo0P0004Coool20000G?oo o`80000Foooo000Joooo0P0005_oool20000?oo o`80001Noooo0P00087oool20000IOooo`80000loooo0P0006Koool20000 4?ooo`004oooo`80001Noooo0P0003Soool20000Goooo`80001ooooo0P00 06Koool20000??ooo`80001Woooo0P0000ooool001?oool20000G_ooo`80 000hoooo0P0005ooool20000Ooooo`80001Voooo0P0003coool20000Iooo o`80000?oooo000Boooo0`0005koool20000>?ooo`80001Ooooo0`0001Ko oom@00005oooo`<0001Voooo0`0003_oool20000Ioooo`<0000>oooo000B oooo0P0005ooool20000>?ooo`<0001Ooooo0P0001Kooom@00005oooo`80 001Xoooo0P0003_oool20000J?ooo`80000>oooo000Boooo0P0005ooool2 0000>Oooo`80001Ooooo0P0007goool20000J?ooo`80000joooo0P0006Wo ool200003_ooo`004Oooo`80001Ooooo0P0003[oool20000H?ooo`80001k oooo0P0006Woool20000>_ooo`80001Zoooo0P0000goool0017oool20000 Goooo`80000joooo0P00063oool20000Noooo`80001Yoooo0P0003[oool2 0000J_ooo`80000=oooo000@oooo0`0005ooool20000>_ooo`80001Poooo 0`0007Woool30000JOooo`80000joooo0P0006[oool300003?ooo`004?oo o`80001Poooo0P0003[oool20000HOooo`80001ioooo0P0006[oool20000 >_ooo`80001[oooo0P0000coool0013oool20000H?ooo`80000joooo0P00 067oool20000NOooo`80001Zoooo0P0003[oool20000Joooo`80000_ooo`80001Roooo0P0007Ooool20000Jooo o`80000joooo0P0006coool200002oooo`003oooo`80001Qoooo0P0003[o ool20000H_ooo`80001goooo0P0006_oool20000>_ooo`80001/oooo0P00 00_oool000koool30000HOooo`80000joooo0P0006;oool30000MOooo`<0 001[oooo0P0003[oool20000K?ooo`<0000:oooo000?oooo0P0005;oool4 00002oooo`80000joooo0P0000[oool40000E?ooo`80001goooo0P0006_o ool20000>_ooo`80001/oooo0P0000_oool000ooool20000D?ooo`L0000: oooo0P0003[oool200002oooo`80001Eoooo0P0007Ooool20000Joooo`80 000joooo0P0006coool200002oooo`004?ooo`80001?oooo0`0000Coool0 0`000?ooooooo`07oooo0P0003[oool200002oooo`80001Doooo0P0007Wo ool20000J_ooo`80000joooo0P0006_oool200003?ooo`004?ooo`80001> oooo0`0000ooool20000>_ooo`80000;oooo0P0005Coool20000NOooo`80 001Zoooo0P0003[oool20000Joooo`80000_ooo`80001Zoooo0`0000coool0017oool20000COooo`80000@ oooo0P0003[oool200002oooo`80001Coooo0P0007_oool20000JOooo`80 000joooo0P0006[oool200003Oooo`004Oooo`80001=oooo0P00013oool2 0000>_ooo`80000;oooo0P0005?oool20000Noooo`80001Yoooo0P0003[o ool20000J_ooo`80000=oooo000Boooo0P0004coool900002Oooo`<0000i oooo0P0000_oool20000D_ooo`80001moooo0P0006Soool20000>_ooo`80 001Yoooo0P0000koool001;oool20000C?ooo`800005oooo0P0000[oool2 0000>Oooo`80000;oooo0P0005;oool20000OOooo`80001Xoooo0P0003[o ool30000J?ooo`80000>oooo000Boooo0`0004coool00`000?ooooooo`03 oooo0P0000[oool20000>?ooo`80000oooo`80001Woooo0`0000koool001?oool2 0000C?ooo`800003oooo0P0000_oool20000>?ooo`80000oooo0P0007ooool20000I_ooo`80000loooo0P0006Ooool200003ooo o`005?ooo`80001Moooo0P0003Soool200003?ooo`80001@oooo0P00087o ool20000IOooo`80000loooo0P0006Koool200004?ooo`005?ooo`80001M oooo0P0003Soool200003?ooo`80001@oooo0P00087oool20000IOooo`80 000loooo0P0006Koool200004?ooo`005?ooo`<0001Loooo0`0003Ooool2 00003?ooo`80001?oooo0`00087oool30000I?ooo`80000loooo0P0006Go ool300004?ooo`005Oooo`80001Moooo0P0003Ooool200003Oooo`040000 oooo00000000C?ooo`800023oooo0P0006Coool20000??ooo`<0001Toooo 0P00017oool001Goool20000GOooo`80000foooo0P0000ooool30000C?oo o`800023oooo0P0006Coool20000?Oooo`80001Toooo0P00017oool001Ko ool20000G?ooo`80000foooo0P0005goool20000QOooo`80001Roooo0P00 03koool20000Hoooo`80000Boooo000Foooo0P0005coool30000=Oooo`80 001Moooo0P0008Goool20000H_ooo`80000noooo0P0006?oool200004_oo o`005_ooo`<0001Loooo0P0003Goool20000G?ooo`<00025oooo0`00067o ool20000?_ooo`<0001Qoooo0`0001;oool001Ooool20000G?ooo`80000d oooo0P0005goool20000Qoooo`80001Qoooo0P0003ooool20000HOooo`80 000Coooo000Goooo0P0005coool20000=?ooo`80001Moooo0P0008Ooool2 0000H?ooo`800010oooo0P00067oool200004oooo`006?ooo`80001Koooo 0`0003?oool20000G?ooo`800029oooo0P0005ooool20000@?ooo`80001P oooo0P0001Coool001Soool20000G?ooo`80000coooo0P0005coool20000 ROooo`80001Ooooo0P00043oool30000Goooo`80000Doooo000Hoooo0`00 05_oool20000<_ooo`80001Loooo0`0008Woool30000G_ooo`800011oooo 0P0005koool300005?ooo`006Oooo`80001Koooo0`00037oool20000G?oo o`80002;oooo0P0005goool20000@_ooo`80001Noooo0P0001Goool001Wo ool20000G?ooo`80000aoooo0P0005coool20000Roooo`80001Moooo0P00 04;oool30000GOooo`80000Eoooo000Joooo0P0005_oool20000?ooo`80000Toooo000Xoooo0P00 06Coool30000J?ooo`80001?oooo2P00053oool20000=oooo`<0001_oooo 0`0003Ooool200009?ooo`00:?ooo`<0001Qoooo100006Soool30000ZOoo o`<0000eoooo0`00077oool30000=Oooo`<0000Toooo000Yoooo0P00063o ool30000J_ooo`80002[oooo0P0003Coool30000Loooo`<0000doooo0P00 02Goool002Woool20000G_ooo`@0000"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{403, 126.063}, ImageMargins->{{0, 0}, {0, 1}}, ImageRegion->{{0, 1}, {0, 1}}]}]}]], "Text", Evaluatable->False, FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[BoxData[ \(RawBracket[ X[b, e, f, a] X[f, e, c, d]] \[Equal] \[Delta][a\ d] \[Delta][ b\ c]\)], "Input"], Cell[BoxData[ \(A\ B\ \[Delta][b\ c]\ \[Delta][ a\ d] + \((A\^2 + B\^2 + A\ B\ dd)\)\ \[Delta][a\ b]\ \[Delta][ c\ d] == \[Delta][b\ c]\ \[Delta][a\ d]\)], "Output"] }, Open ]], Cell[TextData[{ "Thus we see that for the second Reidemeister move to hold, we must have ", Cell[BoxData[ FormBox[ RowBox[{\(A\ B\), "\[Equal]", StyleBox["1", FontFamily->"Courier New"]}], TraditionalForm]]], " and ", Cell[BoxData[ \(A\^2 + B\^2 + A\ B\ d \[Equal] 0\)], "Output"], ". Solving for ", Cell[BoxData[ \(TraditionalForm\`B\)]], " and ", Cell[BoxData[ \(TraditionalForm\`dd\)], FontFamily->"Courier New"], " in terms of ", Cell[BoxData[ \(TraditionalForm\`A\)]], " we get:" }], "Text", Evaluatable->False, CellBaseline->Baseline, AllowInlineCells->True], Cell[CellGroupData[{ Cell[BoxData[ \(rule4\ = \ {B\ -> \ \ 1/A, \ dd\ -> \ \(-A^2\) - 1/A^2}\)], "Input"], Cell[BoxData[ \({B \[Rule] 1\/A, dd \[Rule] \(-\(1\/A\^2\)\) - A\^2}\)], "Output"] }, Open ]], Cell[BoxData[{\(Time\ to\ check\ the\ third\ Reidemeister\ \(move : \ \((It' s\ guaranteed\ to\ hold, \ but\ loose\ nothing\ by\ checking\ again)\)\)\), "\ \[IndentingNewLine]", Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgR00013AP00`40o`00 0?l0003oooooooooooooooooYOooo`00ooooooooooooooooYOooo`00oooo ooooooooooooYOooo`00ooooooooooooooooYOooo`00oooooooooooooooo YOooo`00ooooooooooooooooYOooo`00K?ooo`@000000oooo`000000002g oooo10000?oooom7oooo10000003oooo000000000;Ooool40000K?ooo`00 J_ooo`<00002oooo0`000;Koool70000ooooodCoool300000_ooo`<0002f oooo1`0006_oool006[oool200001?ooo`80002foooo0`0000Coool00`00 0?ooooooo`3ooooo@Oooo`800004oooo0P000;Koool300001?ooo`030000 oooooooo06Soool006Woool200001Oooo`80002eoooo0`000?oooom8oooo 0P0000Goool20000]Oooo`<0001`oooo001Yoooo0P0000Goool20000]Ooo o`80003oooooBOooo`800005oooo0P000;Goool20000LOooo`00JOooo`80 0005oooo0P000;Goool20000ooooodWoool200001Oooo`80002eoooo0P00 077oool006Woool200001Oooo`80002eoooo0P000?oooom9oooo0P0000Go ool20000]Oooo`80001aoooo001Yoooo0P0000Goool20000]Oooo`T0003o oooo@_ooo`800005oooo0P000;Goool90000J_ooo`00JOooo`800005oooo 0P000;Goool200001Oooo`80003ooooo@_ooo`800005oooo0P000;Goool2 00001Oooo`80001Zoooo000[oooo00<0003oooooool09_ooo`80000Doooo 0P0000Coool20000]_ooo`030000oooooooo00?oool200004oooo`030000 oooooooo02Koool20000aOooo`030000oooooooo02Koool200005?ooo`80 0004oooo0P000;Koool00`000?ooooooo`03oooo0P0001?oool00`000?oo ooooo`0Voooo0P0002coool002_oool200009oooo`<0000Coooo0`0000;o ool30000]_ooo`800003oooo0P0001?oool300009oooo`800035oooo0P00 02Ooool300004oooo`<00002oooo0`000;Koool200000oooo`80000Coooo 0`0002Ooool20000;?ooo`00:oooo`80000Xoooo0`0001Coool3000000?o ool000000000^?ooo`@0000Coooo0`0002Soool20000aOooo`80000Xoooo 0`0001Coool3000000?oool000000000^?ooo`@0000Coooo0`0002Soool2 0000;?ooo`00:_ooo`<0000Yoooo0`0001Ooool20000c_ooo`<0000Yoooo 0`000?ooo`<0003Aoooo0`0003Soool20000_Oooo`80000h oooo0`000=7oool30000>?ooo`80000Xoooo000Voooo0`0003Woool30000 coooo`<0000ioooo0`000;_oool30000>Oooo`<0003?oooo0`0003Woool3 00009oooo`009_ooo`80000koooo0`000oooo`80002koooo 0P0003_oool30000cOooo`<0000koooo0P0002Ooool002Koool20000??oo o`<0003;oooo0`0003coool20000^oooo`80000loooo0`000<_oool30000 ??ooo`80000Woooo000Uoooo0`0003goool30000bOooo`<0000moooo0`00 0;Woool30000?Oooo`<00039oooo0`0003goool300009_ooo`009Oooo`80 000ooooo0`000oooo0P0009Woool20000P?ooo`<0001Uoooo0`00083oool20000 5_ooo`005?ooo`<0001_ooo`80002=oooo0P0009So ool30000@Oooo`<0002Hoooo0P00013oool000koool30000>Oooo`<00015 oooo0P0000Coool200004?ooo`<0000ooooo0`00017oool20000B_ooo`<0 000ioooo0`0008_oool30000VOooo`<0000ooooo0`0009Woool300003ooo o`003_ooo`80000ioooo0`0004Koool200001?ooo`80000Aoooo0`0003go ool300004_ooo`80001;oooo0`0003Woool20000Roooo`80002Koooo0`00 03goool30000Voooo`80000?oooo000>oooo0P0003Soool30000Aoooo`80 0004oooo0P0001;oool30000>oooo`<0000Coooo0P0004coool30000>?oo o`80002;oooo0P0009coool30000>oooo`<0002Loooo0P0000ooool000go ool30000=oooo`<00018oooo0P0000Coool200004oooo`<0000ioooo0`00 01Coool20000COooo`<0000goooo0`0008Woool30000WOooo`<0000ioooo 0`0009goool300003_ooo`003Oooo`80000goooo0`0004[oool200000_oo o`80000Eoooo0`0003Ooool300005Oooo`80001>oooo0`0003Ooool20000 ROooo`80002Ooooo0`0003Ooool30000Woooo`80000>oooo000=oooo0P00 03Goool40000C?ooo`L0000Doooo0`0003Goool300005Oooo`<0001?oooo 100003Goool20000ROooo`80002Poooo0`0003Goool30000X?ooo`80000> oooo000oooo 0`0007ooool20000oooo 0P0001koool30000;oooo`<0000@oooo1@0005Woool30000oooo000=oooo0P0009ooool30000=oooo`<0002O oooo0P0008Woool20000=_ooo`@00019oooo0P0001[oool30000=oooo`<0 000;oooo0P0000Coool20000D_ooo`<0000goooo0P0000koool000goool3 0000WOooo`<0000ioooo0`0009goool30000ROooo`<0000goooo0`0004So ool200006Oooo`<0000ioooo0`0000[oool200001?ooo`80001Aoooo0`00 03Ooool300003_ooo`003_ooo`80002Loooo0`0003_oool30000W?ooo`80 0012oooo100004Goool20000>?ooo`<00017oooo0P0001Soool30000>ooo o`<0000:oooo0P0000;oool20000DOooo`<0000hoooo0P0000ooool000ko ool20000Voooo`<0000moooo0`0009_oool20000@_ooo`@00015oooo0P00 03Woool30000AOooo`<0000Goooo0`0003goool300002_ooo`L0001>oooo 0`0003Woool200003oooo`003_ooo`<0002Ioooo0`0003ooool30000VOoo o`<00012oooo100004Goool30000>Oooo`<0001Moooo0`0003ooool30000 GOooo`<0000ioooo0`0000ooool000ooool20000V?ooo`<00011oooo0`00 09Soool20000@oooo`@00016oooo0P0003[oool30000Foooo`<00011oooo 0`0005_oool30000>_ooo`80000@oooo000?oooo0P0009Ooool30000@ooo o`<0002Goooo0P0008goool20000>oooo`@0001Hoooo0`0004?oool30000 F?ooo`@0000koooo0P00013oool000ooool30000UOooo`<00015oooo0`00 09Goool30000SOooo`<0000koooo1000047oool200004oooo`<00015oooo 0`0005Koool30000??ooo`<0000@oooo000@oooo0P0009Coool30000Aooo o`<0002Doooo0P0008ooool20000?Oooo`<00010oooo0P0001;oool30000 Aoooo`<0001Doooo0`0003goool200004Oooo`004?ooo`80002Coooo0`00 04Woool30000Toooo`80002?oooo0P0003koool40000DOooo`<00019oooo 0`00057oool40000?_ooo`80000Aoooo000@oooo0`00097oool30000Booo o`<0002Aoooo0`0004Koool20000Aoooo`<0000noooo100004ooool30000 Boooo`<0001?oooo0`0003ooool300004Oooo`004Oooo`80002@oooo0`00 04goool30000T?ooo`800017oooo0P0004Soool20000@?ooo`<0001=oooo 0`0004goool30000COooo`<00010oooo0P0001;oool0017oool20000Sooo o`<0001?oooo0`0008ooool20000Aoooo`800018oooo0P00047oool40000 B_ooo`<0001?oooo0`0004[oool40000@Oooo`80000Boooo000Aoooo0`00 08goool30000DOooo`<0002=oooo0`0004Ooool20000B?ooo`<00011oooo 100004Soool30000DOooo`<00018oooo0`0004;oool300004_ooo`004_oo o`80002oooo0P0001Koool001Goool20000 Ooooo`<0001Woooo0`0007ooool20000Coooo`H00014oooo0P0004ooool4 0000;?ooo`<0001Woooo0`0002coool40000Coooo`80000Foooo000Eoooo 0`0007goool30000JOooo`<0001moooo0`0004ooool60000A?ooo`<0001? oooo1@0002Woool30000JOooo`<0000Yoooo1000053oool300005_ooo`00 5_ooo`80001loooo0`0006_oool30000O?ooo`800016oooo100000Soool4 0000AOooo`80001Aoooo1@0002Koool30000Joooo`<0000Voooo100005;o ool200005oooo`005_ooo`80001koooo0`0006goool30000Noooo`800016 oooo100000Soool40000AOooo`80001Coooo100002Coool30000KOooo`<0 000Toooo0`0005Coool200005oooo`005_ooo`<0001ioooo0`0006ooool3 0000NOooo`<00016oooo100000Soool40000AOooo`<0001Doooo1000027o ool30000Koooo`<0000Qoooo100005Coool300005oooo`005oooo`80001h oooo0`00077oool30000N?ooo`800017oooo100000Soool40000A_ooo`80 001Eoooo1@0001koool30000LOooo`<0000Noooo100005Koool200006?oo o`005oooo`80001goooo0`0007?oool30000Moooo`800017oooo0P0000[o ool40000A_ooo`80001Goooo0`0001goool30000Loooo`<0000Noooo00<0 003oooooool0E_ooo`80000Hoooo000Goooo0`0007Goool30000MOooo`<0 001eoooo0`0004Ooool200002_ooo`@00016oooo0`0005Soool00`000?oo ooooo`0Joooo0`0007Goool30000MOooo`<0000Hoooo000Hoooo0P0007Co ool30000Moooo`<0001doooo0P0004Soool400001_ooo`@00019oooo0P00 07Coool30000Moooo`<0001doooo0P0001Woool001Soool20000Loooo`<0 001ioooo0`0007?oool20000B?ooo`@00006oooo100004Woool20000Looo o`<0001ioooo0`0007?oool200006Oooo`006?ooo`<0001aoooo0`0007_o ool30000LOooo`<0001:oooo2P0004_oool30000LOooo`<0001koooo0`00 077oool300006Oooo`006Oooo`80001`oooo0`0007goool30000L?ooo`80 001;oooo2P0004coool20000L?ooo`<0001moooo0`00073oool200006_oo o`006Oooo`80001_oooo0`0007ooool30000Koooo`80002Qoooo0P0006oo ool30000Ooooo`<0001_oooo0P0001[oool001Woool30000KOooo`<00021 oooo0`0006goool30000XOooo`<0001]oooo0`00087oool30000KOooo`<0 000Joooo000Joooo0P0006coool30000Poooo`<0001/oooo0P000:?oool2 0000K?ooo`<00023oooo0`0006coool200006oooo`006_ooo`80001[oooo 0`0008Goool30000Joooo`80002Soooo0P0006_oool30000QOooo`<0001[ oooo0P0001_oool001[oool30000JOooo`<00027oooo0`0006Woool30000 Xoooo`<0001Yoooo0`0008Ooool30000JOooo`<0000Koooo000Koooo0P00 06Soool30000ROooo`<0001Xoooo0P000:Goool20000J?ooo`<00029oooo 0`0006Soool200007?ooo`006oooo`80001Woooo0`0008_oool30000Iooo o`80002Uoooo0P0006Ooool30000Roooo`<0001Woooo0P0001coool001_o ool30000IOooo`<0002=oooo0`0006Goool30000YOooo`<0001Uoooo0`00 08goool30000IOooo`<0000Loooo000Loooo0P0006Coool30000Soooo`<0 001Toooo0P000:Ooool20000I?ooo`<0002?oooo0`0006Coool200007Ooo o`007?ooo`80001Soooo0`00097oool30000Hoooo`80002Woooo0P0006?o ool300006Oooo`80001Koooo0`0001Soool30000Hoooo`80000Moooo000L oooo0`00067oool30000Toooo`<0001Qoooo0`000:Ooool30000HOooo`<0 000Joooo1P0005?oool700006Oooo`<0001Qoooo0`0001goool001goool2 0000H?ooo`<0002Eoooo0`00063oool20000ZOooo`80001Poooo0`0001co ool:0000BOooo`T0000Moooo0`00063oool200007_ooo`007Oooo`80001O oooo0`0009Ooool30000Goooo`80002Yoooo0P0005ooool300008Oooo``0 000moooo2`0002;oool30000Goooo`80000Noooo000Moooo0`0005goool3 0000VOooo`<0001Moooo0`000:Woool30000GOooo`<0000Woooo3P0002oo ool=0000:?ooo`<0001Moooo0`0001koool001koool20000G?ooo`<0002K oooo0`0005coool20000Zoooo`80001Loooo0`0002kooolB00006ooooa40 000_oooo0`0005coool200007oooo`007_ooo`80001Koooo0`0009goool3 0000Foooo`80002[oooo0P0005_oool30000=_oooc00000goooo0`0005_o ool200007oooo`007_ooo`<0001Ioooo0`0009ooool30000FOooo`<0002[ oooo0`0005Woool30000@Ooooa`00012oooo0`0005Woool300007oooo`00 7oooo`80001Hoooo0`000:7oool30000F?ooo`80002]oooo0P0005Soool3 0000XOooo`<0001Hoooo0P00023oool001ooool20000Eoooo`<0002Soooo 0`0005Ooool20000[Oooo`80001Goooo0`000:?oool30000Eoooo`80000P oooo000Ooooo0`0005Goool30000YOooo`<0001Eoooo0`000:goool30000 EOooo`<0002Uoooo0`0005Goool300008?ooo`008?ooo`80001Doooo0`00 0:Ooool30000E?ooo`80002_oooo0P0005Coool30000Yoooo`<0001Doooo 0P00027oool0023oool20000Doooo`<0002Yoooo0`0005?oool20000[ooo o`80001Coooo0`000:Woool30000Doooo`80000Qoooo000Poooo0`00057o ool30000Zoooo`<0001Aoooo0`000:ooool30000DOooo`<0002[oooo0`00 057oool300008Oooo`008Oooo`80001@oooo0`000:goool30000D?ooo`80 002aoooo0P00053oool30000[Oooo`<0001@oooo0P0002;oool0027oool2 0000Coooo`<0002_oooo0`0004ooool20000/Oooo`80001?oooo0`000:oo ool30000Coooo`80000Roooo000Qoooo0`0004goool30000/Oooo`<0001= oooo0`000;7oool30000COooo`<0002aoooo0`0004goool300008_ooo`00 8_ooo`80001oooo`<0001Noooo 0P0006goool30000>oooo`80000Woooo000Voooo0`0003Woool30000cooo o`<0000ioooo0`000;_oool30000>Oooo`<0001Ooooo0P0006koool30000 >Oooo`<0000Woooo000Woooo0P0003Soool30000dOooo`<0000hoooo0P00 0;goool20000>?ooo`<0001Poooo0P0006ooool30000>?ooo`80000Xoooo 000Woooo0P0003Ooool30000doooo`<0000goooo0P000;goool20000=ooo o`<0001Poooo0`00073oool30000=oooo`80000Xoooo000Woooo0`0003Go ool30000eOooo`<0000eoooo0`000;goool30000=Oooo`<0003Eoooo0`00 03Goool30000:?ooo`00:?ooo`80000doooo0`000=Ooool30000=?ooo`80 002ooooo0P0003Coool30000eoooo`<0000doooo0P0002Woool002Soool2 0000oooo0P0000Goool20000]?ooo`80 0004oooo0P0000koool30000;Oooo`<0000Zoooo000Zoooo0P0002coool3 00003oooo`800005oooo0P000;Coool200001?ooo`80000?oooo0`0002co ool20000`oooo`80000/oooo0`0000ooool200001Oooo`80002doooo0P00 00Coool200003oooo`<0000/oooo0P0002_oool002[oool20000:oooo`<0 000@oooo0P0000Goool20000]Oooo`800003oooo0P00013oool30000:ooo o`800033oooo0P0002_oool300004?ooo`800005oooo0P000;Goool20000 0oooo`80000@oooo0`0002_oool20000:oooo`00:_ooo`<0000Yoooo0`00 017oool200001Oooo`80002foooo0`000003oooo00000000017oool30000 :Oooo`<00033oooo0`0002Woool300004Oooo`800005oooo0P000;Koool3 000000?oool0000000004Oooo`<0000Yoooo0`0002_oool002_oool20000 :?ooo`<0000Boooo0P0000Goool20000^?ooo`@0000Boooo0`0002Soool2 0000aOooo`80000Xoooo0`0001;oool200001Oooo`80002hoooo100001;o ool30000:?ooo`80000/oooo000[oooo0P0002Ooool300004oooo`800005 oooo0P000;[oool200004oooo`<0000Woooo0P000"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{482, 139.375}, ImageMargins->{{87, 0}, {0, 4}}, ImageRegion->{{0, 1}, {0, 1}}]}], "Text", Evaluatable->False, FontFamily->"Times New Roman"], Cell[BoxData[ \(Bracket[t_]\ := \ Simplify[RawBracket[t]/dd\ /. \ rule4]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Bracket[X[a, b, g, i] X[g, c, d, h] X[i, h, e, f]]\)], "Input"], Cell[BoxData[ \(\(-\(\(1\/\(1 + A\^4\)\)\((A\ \((\[Delta][a\ b]\ \[Delta][d\ e]\ \[Delta][ c\ f] + \[Delta][ c\ d]\ \((A\^4\ \[Delta][b\ e]\ \[Delta][a\ f] + A\^2\ \[Delta][a\ b]\ \[Delta][e\ f])\) + \[Delta][ b\ c]\ \((A\^2\ \[Delta][d\ e]\ \[Delta][a\ f] + \[Delta][ a\ d]\ \[Delta][e\ f])\))\))\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Bracket[X[b, c, i, h] X[d, e, g, i] X[g, f, a, h]]\)], "Input"], Cell[BoxData[ \(\(-\(\(1\/\(1 + A\^4\)\)\((A\ \((\[Delta][a\ b]\ \[Delta][d\ e]\ \[Delta][ c\ f] + \[Delta][ c\ d]\ \((A\^4\ \[Delta][b\ e]\ \[Delta][a\ f] + A\^2\ \[Delta][a\ b]\ \[Delta][e\ f])\) + \[Delta][ b\ c]\ \((A\^2\ \[Delta][d\ e]\ \[Delta][a\ f] + \[Delta][ a\ d]\ \[Delta][e\ f])\))\))\)\)\)\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ "The", " ", "remaining", " ", "sad", " ", "point", " ", "is", " ", "the", " ", "behavior", " ", "of", " ", "the", " ", "bracket", " ", "under", " ", "the", " ", "first", " ", "Reidemeister", " ", RowBox[{ "move", ":", "\[IndentingNewLine]", Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooo0P00 0?ooo`80000Poooo 000Roooo0P0002;oool20000IOooo`80002Soooo0P0002goool20000>?oo o`80000Poooo000Qoooo0`0002;oool20000IOooo`<0002Qoooo0`0002go ool20000>?ooo`<0000Ooooo000Qoooo0P0002?oool20000I_ooo`80002Q oooo0P0002koool20000>Oooo`80000Ooooo000Qoooo0P0002?oool20000 I_ooo`80002Qoooo0P0002koool20000>Oooo`80000Ooooo000Poooo0P00 02Coool20000Ioooo`80002Ooooo0P0002ooool20000>_ooo`80000Noooo 000Poooo0P0002Coool20000Ioooo`80002Ooooo0P0002ooool20000>_oo o`80000Noooo000Ooooo0`0002Coool20000Ioooo`<0002Moooo0`0002oo ool20000>_ooo`<0000Moooo000Ooooo0P0002Goool20000J?ooo`80002M oooo0P00033oool20000>oooo`80000Moooo000Ooooo0P0002Goool20000 J?ooo`80002Moooo0P00033oool20000>oooo`80000Moooo000Noooo0P00 02Koool20000JOooo`80002Koooo0P00037oool20000??ooo`80000Loooo 000Noooo0P0002Koool20000JOooo`80002Koooo0P00037oool20000??oo o`80000Loooo000Moooo0`0002Koool20000JOooo`<0002Ioooo0`00037o ool20000??ooo`<0000Koooo000Moooo0P0002Koool30000J_ooo`80002I oooo0P0003;oool20000?Oooo`80000Koooo000Moooo0P0002Koool20000 Joooo`80002Ioooo0P0003;oool20000?Oooo`80000Koooo000Loooo0P00 02Ooool20000?ooo`800013oooo0P0001Goool001Koool20000 ;_ooo`80000Koooo0`0002[oool200009oooo`80002;oooo0P0003Woool2 0000A?ooo`80000Doooo000Foooo0P0002koool200006oooo`80000/oooo 0P0002Koool20000Roooo`80000ioooo0P0004Coool200005?ooo`005Ooo o`<0000^oooo0P0001[oool20000;Oooo`<0000Uoooo0`0008Woool30000 >Oooo`800014oooo0`0001?oool001Goool20000;oooo`80000Ioooo0`00 02koool300009Oooo`800029oooo0P0003[oool20000AOooo`80000Coooo 000Eoooo0P0002ooool200006Oooo`80000`oooo0`0002Coool20000ROoo o`80000joooo0P0004Goool200004oooo`005?ooo`80000`oooo0`0001Oo ool30000oooo`800016oooo0P00 01;oool001Coool20000?ooo`800011oooo100000[oool200009?oo o`80000FooooD00001Ooool20000@?ooo`80001;oooo0P0000goool000oo ool20000>?ooo`80000ooooo1`0000Woool300008oooo`80001moooo0P00 043oool20000Boooo`80000=oooo000>oooo0P0003[oool20000?_ooo`<0 0004oooo00<0003oooooool01oooo`80000Toooo0P0007_oool20000@Ooo o`80001_ooo`80000moooo0`0000oo ool200009?ooo`80001koooo0P00047oool20000C?ooo`80000oooo0P0000[oool000_oool30000?Ooo o`<0000loooo00<0003oooooool00oooo`800009oooo0`0002Goool30000 MOooo`<00013oooo0P0004koool300002Oooo`003?ooo`80000noooo0P00 03coool200000oooo`<00009oooo0P0002Koool20000Moooo`800013oooo 0P0004koool200002_ooo`003?ooo`80000noooo0P0003koool500002_oo o`80000Voooo0P0007Ooool20000@oooo`80001>oooo0P0000[oool000go ool20000?Oooo`80001=oooo0P0002Goool20000NOooo`800012oooo0P00 04goool200002oooo`003Oooo`80000noooo0P0004_oool200009_ooo`80 001ioooo0P0004;oool20000COooo`80000;oooo000=oooo0`0003goool2 0000Boooo`80000Uoooo0`0007Woool30000@Oooo`80001oooo000@oooo0P0003goool2 0000A_ooo`80000Voooo0P0007ooool20000?oooo`80001:oooo0P0000ko ool0017oool20000??ooo`<00014oooo0`0002Goool20000POooo`80000n oooo0P0004Woool200003oooo`004Oooo`80000moooo0P0004Coool20000 9_ooo`800021oooo0P0003koool20000BOooo`80000?oooo000Aoooo0`00 03coool30000@_ooo`80000Voooo0`00087oool30000?Oooo`800018oooo 0`0000ooool001;oool20000?Oooo`800012oooo0P0002Koool20000Pooo o`80000moooo0P0004Soool200004?ooo`004_ooo`80000moooo0`00043o ool200009oooo`800023oooo0P0003goool20000B?ooo`80000@oooo000C oooo0P0003goool20000?oooo`<0000Voooo0P0008Goool20000??ooo`80 0017oooo0P00017oool001?oool20000?Oooo`<0000moooo0`0002Ooool2 0000QOooo`80000loooo0P0004Ooool200004Oooo`004oooo`<0000moooo 0P0003goool200009oooo`<00025oooo0`0003_oool20000A_ooo`<0000A oooo000Doooo0P0003koool20000>oooo`80000Xoooo0P0008Ooool20000 >oooo`800016oooo0P0001;oool001Coool20000?_ooo`80000joooo0`00 02Soool20000Qoooo`80000koooo0P0004Koool200004_ooo`005Oooo`80 000noooo0P0003Woool20000:?ooo`800029oooo0P0003[oool20000AOoo o`80000Coooo000Eoooo0P0003koool30000=oooo`80000Yoooo0P0008Wo ool20000>_ooo`800015oooo0P0001?oool001Goool30000?_ooo`80000f oooo0`0002Soool30000ROooo`<0000ioooo0P0004Coool300004oooo`00 5_ooo`80000ooooo0P0003Coool30000:Oooo`80002;oooo0P0003Woool2 0000A?ooo`80000Doooo000Foooo0P0003ooool30000<_ooo`<0000Zoooo 0P0008_oool20000>Oooo`800014oooo0P0001Coool001Ooool20000?ooo o`<0000`oooo0`0002[oool20000SOooo`80000hoooo0P0004?oool20000 5Oooo`005oooo`800010oooo0P0002ooool30000:oooo`80002=oooo0P00 03Soool20000@oooo`80000Eoooo000Goooo0`0003ooool30000;Oooo`<0 000[oooo0`0008goool30000=oooo`800012oooo0`0001Goool001Soool2 0000@?ooo`<0000[oooo0`0002coool20000A?ooo`@00017oooo0P0003Oo ool20000@_ooo`80000Foooo000Hoooo0P00047oool400009oooo`@0000] oooo0P0004Coool40000Aoooo`80000goooo0P0004;oool200005_ooo`00 6Oooo`800011oooo1@0002Coool30000;_ooo`800015oooo100004Soool2 0000=_ooo`800011oooo0P0001Ooool001Woool20000@oooo`@0000Roooo 0`0002ooool20000AOooo`@00018oooo0P0003Koool20000@Oooo`80000G oooo000Ioooo0`0004Coool400007_ooo`@0000_oooo0`00097oool30000 =Oooo`800010oooo0`0001Ooool001[oool20000A_ooo`L0000Hoooo0`00 037oool20000Toooo`80000eoooo0P00043oool200006?ooo`006_ooo`80 0017oooo200001?oool50000<_ooo`80002Coooo0P0003Goool20000@?oo o`80000Hoooo000Koooo0P0004_oool600003Oooo`L0000boooo0P0009Go ool20000=?ooo`80000ooooo0P0001Woool001_oool20000C_oooa@0000e oooo0P0004Woool20000B_ooo`80000doooo0P0003ooool200006Oooo`00 6oooo`<0001?oooo3`0003Ooool30000BOooo`80001:oooo0`0003?oool2 0000?_ooo`<0000Ioooo000Loooo0P0002goool200009oooo`030000oooo oooo03coool20000B_ooo`80001;oooo0P0003?oool20000?_ooo`80000J oooo000Loooo0P0002goool20000I_ooo`80001:oooo0P0004_oool20000 _ooo`<0000Moooo000Poooo0P0002Woool20000H_ooo`80 001@oooo1P0004Woool20000;oooo`80000joooo0P0001koool0023oool2 0000:Oooo`80001Roooo0P00053oool60000BOooo`80000_oooo0P0003[o ool200007_ooo`008Oooo`80000Xoooo0P00067oool20000Doooo`H00018 oooo0P0002koool20000>Oooo`80000Ooooo000Qoooo0P0002Soool20000 HOooo`80001Coooo1P0004Soool20000;_ooo`80000ioooo0P0001ooool0 027oool300009oooo`80001Poooo0`0004Woool400002?ooo`@00018oooo 0`0002goool20000>?ooo`<0000Ooooo000Roooo0P0002Ooool20000H?oo o`80001:oooo100000Soool40000BOooo`80000]oooo0P0003Soool20000 8?ooo`008_ooo`80000Woooo0P00063oool20000B_ooo`@00008oooo1000 04Woool20000;Oooo`80000hoooo0P00023oool002?oool200009_ooo`80 001Ooooo0P0004_oool400002?ooo`@0001:oooo0P0002coool20000=ooo o`80000Qoooo000Soooo0P0002Koool20000Goooo`80001;oooo0P0000[o ool40000B_ooo`80000/oooo0P0003Ooool200008Oooo`008oooo`<0000U oooo0P0005koool30000Boooo`80000:oooo100004[oool30000:oooo`80 000foooo0`00027oool002Coool200009Oooo`80001Noooo0P0004coool4 00001_ooo`@0001=oooo0P0002_oool20000=_ooo`80000Roooo000Toooo 0P0002Goool20000G_ooo`80001oooo100003goool20000aOooo`80 000Loooo0P0000koool400005Oooo`80000aoooo000coooo0P0001Koool2 00004?ooo`80000moooo0P000"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{271, 126.563}, ImageMargins->{{153, 0}, {0, 7}}, ImageRegion->{{0, 1}, {0, 1}}]}]}]], "Text", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[RawBracket[X[c, a, b, c]]\ /. \ rule4]\)], "Input"], Cell[BoxData[ \(\(-A\^3\)\ \[Delta][a\ b]\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{\(This\ is\ unfortunate\), ",", " ", RowBox[{ "but", " ", "not", " ", \(fatal . \ It\), " ", "can", " ", "be", " ", "fixed", " ", "by", " ", "multiplying", " ", "the", " ", "bracket", " ", "by", " ", Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(-A\^3\)\)]]]], " ", "raised", " ", "to", " ", "an", " ", "appropriate", " ", "multiple", " ", "of", " ", "the", " ", \(writhe . \ It\), " ", "is", " ", "common", " ", "to", " ", "make", " ", "the", " ", "further", " ", "substiturion", " ", Cell[ TextData[Cell[BoxData[ \(TraditionalForm\`A \[Rule] q\^\(1/4\)\)]]]], " ", "and", " ", "then", " ", "the", " ", "resulting", " ", "invariant", " ", "is", " ", "called", " ", \("the Jones polynomial" . \ Let\), " ", "us", " ", "compute", " ", "it", " ", "for", " ", "the", " ", "trefoil", " ", "and", " ", "its", " ", "mirror", " ", \(\(image\)\(:\)\)}]}]], "Text", PageWidth->WindowWidth, Evaluatable->False, TextJustification->1, FontFamily->"Times New Roman"], Cell[BoxData[ \(Jones[t_, \ w_]\ := \ Simplify[\((Bracket[t]\ \((\(-A^3\))\)^\((\(-w\))\))\)\ /. \ A \[Rule] q^\((1/4)\)]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Jones[Trefoil, \ 3]\)], "Input"], Cell[BoxData[ \(\(\(-1\) + q + q\^3\)\/q\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MirrorTrefoil\ = \ \ Trefoil\ /. \ x_X\ \[RuleDelayed] \ RotateLeft[x]\)], "Input"], Cell[BoxData[ \(X[b, f, c, e]\ X[d, b, e, a]\ X[f, d, a, c]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Jones[MirrorTrefoil, \(-3\)]\)], "Input"], Cell[BoxData[ \(q + q\^3 - q\^4\)], "Output"] }, Open ]], Cell["Ok, if you insist,", "Text"], Cell[BoxData[ \(w[K_]\ := \ Module[\[IndentingNewLine]{p, l}, \[IndentingNewLine]l = ReplacePart[ K /. \[InvisibleSpace]X[a_, b_, c_, d_] \[RuleDelayed] \[Delta][a\ c]\ \[Delta][b\ d], p, {1, 0}] //. \[InvisibleSpace]{p[a_\ b_] \[RuleDelayed] p[a, b], p[a_, b___, c_]\ \[Delta][c_\ d_] \[RuleDelayed] p[a, b, c, d]}; \[IndentingNewLine]l = MapThread[ Rule, {List @@ l, Range[Length[ l]]}]; \[IndentingNewLine]\((Plus\ @@ \ \((K\ /. \ l)\))\)\ /. \ X[a_, b_, c_, d_]\ \[RuleDelayed] \ If[Abs[p = \((a - c)\) \((b - d)\)] \[Equal] 1, p, \(-Sign[p]\)]\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(Jones[K_]\ := \ Jones[K, w[K]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \({Jones[Trefoil], Jones[MirrorTrefoil]}\)], "Input"], Cell[BoxData[ \({\(\(-1\) + q + q\^3\)\/q\^4, q + q\^3 - q\^4}\)], "Output"] }, Open ]], Cell["Now, last but not least,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((\[IndentingNewLine]q*\((\(-A^3\))\)^\((\(-1\))\)* RawBracket[X[a, b, c, d]] - q^\((\(-1\))\)*\((\(-A^3\))\)* RawBracket[ X[b, c, d, a]]\[IndentingNewLine])\)\ /. \ {A \[Rule] q^\((1/4)\), \ B \[Rule] q^\((\(-1\)/4)\)}\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(\(\[Delta][b\ c]\ \[Delta][a\ d]\)\/\@q - \@q\ \[Delta][b\ c]\ \[Delta][ a\ d]\)], "Output"] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1920}, {0, 1342}}, WindowToolbars->"EditBar", WindowSize->{1082, 1310}, WindowMargins->{{170, Automatic}, {2, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PrintingMargins"->{{54, 54}, {72, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->True, "PrintMultipleHorizontalPages"->False}, CellLabelAutoDelete->True, Magnification->1.5, StyleDefinitions -> "Default.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1717, 49, 21214, 350, 325, "Text"], Cell[CellGroupData[{ Cell[22956, 403, 90, 1, 42, "Input"], Cell[23049, 406, 77, 1, 41, "Output"] }, Open ]], Cell[23141, 410, 39348, 653, 372, "Text", Evaluatable->False], Cell[62492, 1065, 180, 3, 42, "Input"], Cell[62675, 1070, 114, 3, 47, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[62814, 1077, 53, 1, 42, "Input"], Cell[62870, 1080, 326, 6, 66, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[63233, 1091, 67, 1, 42, "Input"], Cell[63303, 1094, 921, 15, 121, "Output"] }, Open ]], Cell[64239, 1112, 4481, 78, 113, "Text", Evaluatable->False], Cell[68723, 1192, 126, 2, 42, "Input"], Cell[CellGroupData[{ Cell[68874, 1198, 87, 1, 42, "Input"], Cell[68964, 1201, 387, 7, 69, "Output"] }, Open ]], Cell[69366, 1211, 138, 4, 47, "Text", Evaluatable->False], Cell[69507, 1217, 126, 2, 42, "Input"], Cell[69636, 1221, 79, 1, 47, "Text", Evaluatable->False], Cell[69718, 1224, 144, 3, 42, "Input"], Cell[CellGroupData[{ Cell[69887, 1231, 52, 1, 42, "Input"], Cell[69942, 1234, 107, 2, 43, "Output"] }, Open ]], Cell[70064, 1239, 21072, 350, 264, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[91161, 1593, 131, 3, 42, "Input"], Cell[91295, 1598, 193, 3, 43, "Output"] }, Open ]], Cell[91503, 1604, 657, 24, 73, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[92185, 1632, 91, 1, 42, "Input"], Cell[92279, 1635, 86, 1, 57, "Output"] }, Open ]], Cell[92380, 1639, 31180, 515, 290, "Text", Evaluatable->False], Cell[123563, 2156, 91, 1, 42, "Input"], Cell[CellGroupData[{ Cell[123679, 2161, 83, 1, 42, "Input"], Cell[123765, 2164, 433, 7, 85, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[124235, 2176, 83, 1, 42, "Input"], Cell[124321, 2179, 433, 7, 85, "Output"] }, Open ]], Cell[124769, 2189, 18615, 309, 273, "Text"], Cell[CellGroupData[{ Cell[143409, 2502, 81, 1, 42, "Input"], Cell[143493, 2505, 59, 1, 43, "Output"] }, Open ]], Cell[143567, 2509, 1125, 20, 96, "Text", Evaluatable->False], Cell[144695, 2531, 158, 3, 42, "Input"], Cell[CellGroupData[{ Cell[144878, 2538, 52, 1, 42, "Input"], Cell[144933, 2541, 61, 1, 65, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[145031, 2547, 115, 2, 42, "Input"], Cell[145149, 2551, 77, 1, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[145263, 2557, 61, 1, 42, "Input"], Cell[145327, 2560, 49, 1, 43, "Output"] }, Open ]], Cell[145391, 2564, 34, 0, 47, "Text"], Cell[145428, 2566, 821, 17, 198, "Input"], Cell[146252, 2585, 64, 1, 42, "Input"], Cell[CellGroupData[{ Cell[146341, 2590, 71, 1, 42, "Input"], Cell[146415, 2593, 80, 1, 65, "Output"] }, Open ]], Cell[146510, 2597, 40, 0, 47, "Text"], Cell[CellGroupData[{ Cell[146575, 2601, 362, 7, 120, "Input"], Cell[146940, 2610, 122, 2, 64, "Output"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)