{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading \+ 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE " " -1 -1 "MS Serif" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "C ourier" 1 12 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 45 "Worksheet IV: Quandle Coc ycle Knot Invariants" }}{PARA 19 "" 0 "" {TEXT -1 33 "by Masahico Sait o and Chad Smudde" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 1178 " \+ This worksheet will explain the new procedures created for calculati ng quandle cocycle knot invariants of classical knots. Some of the pro cedure contained in this package are similar to those explained in the previous worksheet, but we have modified these to increase the speed \+ at which the braids are colored. This was the main goal of this update . It is now possible to color braids with much larger quandles( ie. qu andles with 25 elements or much more, over 100 elements, in some circu mstances). The second major upgrade was creating easy to use procedure s that would do mass calculations of these invariants at a minimal cos t to the user. These procedures, once executed, could run for countles s hours at no expense to the user and organize the data in a relativel y efficient and effective manner. This worksheet should be considered \+ a second version and compilation of the three previous worksheets. It \+ is currently only configured to run in Linux. Porting to other O.S.'s \+ is possible and may be simple in some cases. Explanations of why this \+ is Linux specific and changes that should be made to allow use in othe r O.S.'s will be given throughout this worksheet " }}}{EXCHG {PARA 0 " " 0 "" {TEXT 200 594 " A web site was created to help organiz e this project. It can be found at http://shell.cas.usf.edu/quandle/. \+ This web site contains a background section that briefly explains the \+ theory that governs this topic. This worksheet is not intended to give such background information, but rather to explain how to use the pro cedures created to perform these calculations. One may also with to lo ok at the book, ``Surfaces in 4-space,'' by Carter, Kamada, and Saito, Springer-Verlag, 2004. The reference section in the background secti on of the web page also contains many other references. " }}{PARA 0 " " 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 63 " \+ This worksheet is divided into the following sections:" }}{PARA 0 "" 0 "" {TEXT 200 28 " I. Braids" }}{PARA 0 "" 0 "" {TEXT 200 30 " II. Quandles" }}{PARA 0 "" 0 "" {TEXT 200 44 " III. Coloring braids with C" }}{PARA 0 "" 0 " " {TEXT 200 134 " IV. Calculating the value of the inv ariant\n V. Mass calculations with automatic file nam e generation" }}{PARA 0 "" 0 "" {TEXT 200 15 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 300 "First the package containing the procedures will be read into the worksheet. Note: If you place all of the files for this project in the same directory you will not need to specify the paths to them. The procedures contained in this package \+ are listed here and explained in the appropriate section.\n" }}{PARA 0 "" 0 "" {TEXT -1 495 "validColorVectors:=define_external(\n \+ 'validcolorvectorsint',\n 'quandle': :ARRAY(datatype=integer[2]),\n 'quandlesize'::int eger[4],\n 'knot'::ARRAY(datatype=integer[2]),\n \+ 'braidindex'::integer[4],\n \+ 'nopsknot'::integer[4],\n 'validCvecs'::ARRAY(dat atype=integer[2]),\n RETURN::integer[4],\n \+ LIB=\"ValCol8_1_05.so\"):" }}{PARA 0 "" 0 "" {TEXT -1 345 "cvrtarray:=proc(brind,nopknt,L,numVC);\nInvar2_3UntwistedDLL:=pro c(Quandle,p,Knot);\nCalcUntwistinvars:=proc(filePATH,p ,qord,qnum,numb erofknots,\{cpath:=false\},\{solpath=false\});\nfpQuan:=proc(tfile,Qua ndle,qsize);\nInvar2UntwistedDLL:=proc(Quandle,p,Knot);\nCalc2Untwisti nvars:=proc(filePATH,p,qord,qnum,numberofknots,\{cpath=false\},\{solpa th:=false\});" }}{PARA 0 "" 0 "" {TEXT -1 41 "Invar3UntwistedDLL:=proc (Quandle,p,Knot);" }}{PARA 0 "" 0 "" {TEXT -1 311 "Calc3Untwistinvars: =proc(filePATH,p,qord,qnum,numberofknots,\{cpath=false\},\{solpath:=fa lse\});\nInvar2_3MochizukiDLL:=proc(Xpolym,Apolym,p,m1,m2,a3,n1,b2,Kno t);\nCalcMochinvars:=proc(filePATH,p ,polym,m1,m2,a3,n,b,numberofknots ,\{cpath:=false\},\{solpath:=false\});\nInvar2MochizukiDLL:=proc(Xpoly m,Apolym,p,n1,b2,Knot);" }}{PARA 0 "" 0 "" {TEXT -1 91 "Calc2Mochinvar s:=proc(filePATH,p ,polym,n,b,numberofknots,\{cpath:=false\},\{solpath :=false\});" }}{PARA 0 "" 0 "" {TEXT -1 56 "Invar3MochizukiDLL:=proc(X polym,Apolym,p,m1,m2,a3,Knot);" }}{PARA 0 "" 0 "" {TEXT -1 811 "Calc3M ochinvars:=proc(filePATH,p ,polym,m1,m2,a3,numberofknots,\{cpath:=fals e\},\{solpath:=false\});\nismult:=proc(charstar);\nCalcDihinvars:=proc (filePATH,p,numberofKnots,\{cpath:=false\},\{solpath:=false\});\nco2Tw istedSol:=proc(Quandle,polym,p);\nInvar2TwistedDLL(Quandle,knot,polym, p);\nCalc2twistinvars(filePATH,p,polym,qord,qnum,numberofknots,\{cpath :=false\},\{solpath:=false\});\ntinverse(polym,p);\n\nquandlesize:=pro c(Quandle);\nmakeinv:=proc(Quandle);\nco2Solution:=proc(Quandle,m::pos int);\nmirrorKnot:=proc(L);\nbarKnot:=proc(L);\nconnSumKnots:=proc(Kno ts_list);\nco3Solution:=proc(Quandle,m::posint);\nMochizuki3coc:=proc( Xpolym,Apolym,p,m1,m2,a3);\ncocycle3check:=proc(cocy,Xpolym,Apolym,p); \nXi:=proc(n);\nMochizuki2coc:=proc(Xpolym,Apolym,p,m1,a2);\ncocycle2c heck:=proc(cocy,Xpolym,Apolym,p);\nMochizukiDihedral3coc:=proc(p);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "#Change the path to where you stored the \+ package\nread \"/media/sda1/quancocDLLpkgmaple8.m\":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "This define external statement was not properl y saved to the package. You will need to copy and paste it into any wo rksheet that you load the quancocDLLpkgmaple8.m package into. This is \+ vital since most of the procedures depend on this define_external stat ement." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "#Change the path to where you saved the ValCol8_1_05.so file.\n\nvalidColorVectors:=de fine_external(\n 'validcolorvectorsint',\n \+ 'quandle'::ARRAY(datatype=integer[2]),\n \+ 'quandlesize'::integer[4],\n 'knot'::ARRAY (datatype=integer[2]),\n 'braidindex'::integer[4] ,\n 'nopsknot'::integer[4],\n \+ 'validCvecs'::ARRAY(datatype=integer[2]),\n RET URN::integer[4],\n LIB=\"/media/sda1/WorksheetIV/ ValCol8_1_05.so\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "This proc edure was not included into the package. It is used in section 3 of th is worksheet. It is mostly for demonstration purposes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 749 "printColors:=proc(Knot,Col,numVc) \nlocal i,j,k,brind,q,r,ss,threeprint;\n\nbrind:=max(op(map(x->abs(x), Knot)))+1; \nq:=iquo(numVc,3);\nr:=numVc mod 3;\ni:=1;\n for ss from 1 to q do\ni:=ss*3-2;\n for j from 1 to nops(Knot) +1 do\n \+ for threeprint from 1 to 3 do\n for k from 1 to brind do\n \+ printf(\"%5d\",Col[i][j][k] );\n od;\n i:=i +1;\n printf(\"\\t\\t\");\nod;\ni:=i-3;\nprintf(\"\\n\");\nod; \n\nprintf(\"\\n\\n\");\nod;\ni:=ss*3-2;\n#for ss from 1 to r do\n \+ for j from 1 to nops(Knot) +1 do\n for threeprint from 1 to r do\n for k from 1 to brind do\n printf(\"%5d\",C ol[i][j][k] );\n od;\n i:=i+1;\n printf(\"\\t \\t\");\nod;\ni:=i-r;\nprintf(\"\\n\");\nod;\nprintf(\"\\n\\n\");\n#od ;\n\n\n\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 11 "I. Braids" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 609 " Worksheet III was where we saw the chang e in braids. We changed to the table of braids generated form Charles \+ Livingstons's Knotinfo database. This allowed us to increase the numbe r of knots we use to those with 12 and fewer crossings. We also added \+ the knot name and the Alexander polynomial to the data stored for each knot. Printing the knot name with the value of the invariant allows u s to more easily recognize the knot that was calculated. Adding the Al exander polynomial to the data file was motivated by Inoue's theorem. \+ Procedures for braid manipulation are also included in this package. \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 180 " The procedures and files pertaining to braids are:\n knotsLivingston.txt\n mirrorKnot(),\n barKnot(),\n \+ connSumKnots()." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 123 "First read the knot data into the worksheet. \nNote: A colon should be used here to avoid printing the 2977 different knots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "#Alter path here\n\nread \"/media/sda1/Worksheet IV/knotsLivingston.txt\":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 623 "Th e data is stored in a two dimensional array that has the name \"Knot\" . The data is accessed by the command, [> Knot[i,j]; . The first index , i, specifies the knot you wish to access. i ranges from 0 to 2976. T he knots were ordered as they appeared in Charles Livingston's Knotinf o database. The second index, j, ranges from 1 to 3. j=1 will give the knot name which is stored as a string. j=2 will give the braid word. \+ This is stored as a list indexed by one. j=3 will give the Alexander p olynomial of the knot. This was not stored as a string, but as an expr ession so that it will be of type polynom when read into Maple." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 17 "The trefoil knot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Knot[0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q$3_16\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " Knot[0,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Knot[0,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"tG\"\"#\"\"\"F(F&!\"\"F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 200 25 "The final knot available." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "Knot[2976,1];Knot[2976,2];Knot[2976,3];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q)12n_08886\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.!\"\"F$F$!\"#F%F%F$F$F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"tG!\"\"*$)F%\"\"#F$F&*&\"\"'F$)F%\"\"$F$F$ *&\"#6F$)F%\"\"%F$F&*&\"#8F$)F%\"\"&F$F$*&F/F$)F%F+F$F&*&F+F$)F%\"\"(F $F$*$)F%\"\")F$F&*$)F%\"\"*F$F&*$)F%\"#5F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "type(Knot[0,3],polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 184 "Since the Alexander polynomial is of type polynom, we can check the gcd wit h some other polynomial that may be used for the Alexander quandle. Se e Inoue's theorem for more information." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Gcd(Knot[0,3],t^2-t+1) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"tG\"\"#\"\"\"F(F&F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 243 "The mirror image is obtained with the procedure mi rrorKnot(). This procedure will only work on the braid word of the kno t. Any naming must be performed manually. The procedure will return th e braid word for the mirror image of the given braid." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mirrorKnot(Knot[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&!\"#\"\"\"F$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 179 "The braid with orientation reversed can be obtained wit h the procedure, barKnot(). This will also only work on the braid word of the knot. Any naming should be performed manually." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "barKnot(Knot[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&!\"#\"\"\"F$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 158 "The procedure connSumKnots takes a list of lists as is argume nt. The elements of the list are the braid words that are to be connec ted in the specified order." }}{PARA 0 "" 0 "" {TEXT 200 149 "Note: Th is procedure just connects the braids. It does NOT try to reduce the l ength. The new braid word is returned as a single list indexed by one. ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "connSumKnots([Knot[0,2 ],Knot[1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"\"F$F$\"\"#\" \"$!\"%F&F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 171 "Note: special pr ocedures for calculating the braid index or number of crossings of a g iven braid are not included since this information can be obtained fro m simple calls." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 43 "The braid ind ex is recovered from the call." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "brind:=max(op(map(x->abs(x),Knot[0,2])))+1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&brindG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 24 "The number of crossings." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nops(Knot[0,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 15 "II. Quandles" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 366 " There are two different files that w ill give us access to quandles. The first is \"Quandle.\" This is the \+ same file that was used in worksheets I and II (the untwisted and twis ted cases). The second is the package used to generate the Cayley tabl e for certain Alexander Quandles. This is the one used in worksheet II I, the Mochizuki polynomial cocycle case. " }}}{EXCHG {PARA 4 "" 0 " " {TEXT -1 61 "Read the quandles into the worksheet from the file \"Qu andle.\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 658 " The file \+ \"Quandle\" contains all quandles from three to six elements that are \+ pairwise non-isomorphic with non-trivial torsion part in its second or third homology groups. The list is from the book ``Surfaces in 4-spa ce,'' by Carter, Kamada, Saito, Springer-Verlag, 2004. They are labele d Qi_j where i is the size and j is which quandle. There are 25 six el ement, 6 five element, 3 four element, and 1 three element quandles in the file \"Quandle.m\". These are two dimensional zero indexed arrays representing the multiplication table a*b for a,b in X, the quandle. \+ This step could be skipped if you don't want to work with any quandle \+ in this list. " }}{PARA 0 "" 0 "" {TEXT 200 63 "Note: A colon should b e used to avoid printing all of the data." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "#alter path here \n\nread \"/media/sda1/WorksheetIV/Quandle\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print(convert(Q3,matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!\"\"#\"\"\"7%F)F*F(7%F*F(F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "print(convert(Q5_4,matrix)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"!F(F(F(F(7'\" \"\"F*\"\"%\"\"#\"\"$7'F,F-F,F+F*7'F-F+F*F-F,7'F+F,F-F*F+" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 54 "Read the Alexander quandle package into t he worksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#alter pat h here\n\nread \"/media/sda1/WorksheetIV/alexQuanpkg.m\";" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 495 " Recall that Z_p[t,t^-1]/(h(t) ) where h(t)=a_0+a_1*t+...+a_n*t^n is a polynomial in t with quandle o peration a*b=ta+(1-t)b is a finite quandle. The procedure AlexQuandle( ) will generate the Cayley table for this quandle and return an isomor phic quandle whose Cayley table has elements in \{0..n-1\} where n is \+ the order of the quandle. This will allow us to reuse old programs tha t are set to only accept quandles that are zero indexed two dimensiona l arrays with elements in \{0..n-1\}. " }}{PARA 0 "" 0 "" {TEXT 200 968 " Since p is prime and h(t)=a_0+a_1*t+...+a_n*t^n is a po lynomial (in t), elements of Z_p[t,t^-1]/(h(t)) are represented by th e reduced residue system, i.e., polynomials of the form b_0+b_1*t+...+ b_(n-1)*t^(n-1) where b_i is in Z_p. Since p is prime 1..p-1 are units in Z_p, so that we can solve (t^(-1))*h(t)=0 for t^-1. This implies \+ that t^(-1) is expressed by a polynomial as above, and every element i s represented by a polynomial of a positive degree. Then by taking the remainder when divided by h(t), every element is represented by a pol ynomial as above. The purpose of the local procedure rres() is to list all such polynomials. Next, the procedure Alexmult() will multiply al l possibilities a*b where a, b are in the reduced residue system. The \+ resulting polynomial will then be reduced back into the reduced residu e system. Finally, the procedure findelt() will find that polynomial i n the list of reduced residues and return that integer value. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Q9:=AlexQuandle(t^2-t+1,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q9G%(QuandleG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(Q9,matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7+7+\"\"!\"\"(\"\"&\"\"\"\"\")\"\"$\"\"#\" \"'\"\"%7+F-F+F,F0F.F/F*F(F)7+F/F0F.F)F*F(F,F-F+7+F*F(F)F-F+F,F0F.F/7+ F,F-F+F/F0F.F)F*F(7+F.F/F0F(F)F*F+F,F-7+F)F*F(F,F-F+F/F0F.7+F+F,F-F.F/ F0F(F)F*7+F0F.F/F*F(F)F-F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 128 " This procedure can also be used to generate the Dihedral quandles si nce the Dihedral quandle is the special case when h(t)=t+1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R3:=AlexQuandle(t+1,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R3G%(QuandleG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "print(convert(R3,matrix));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!\"\"#\"\"\"7%F)F*F(7%F*F(F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 289 "The following command may be use ful for calculating the order of the quandle. This will avoid the user from specifying the order in the case that the quandle is passed to a procedure. This command is unnecessary for the procedures included he re, but may be useful for procedure you create." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "quandleorder:=linalg[coldim](convert(R3,matrix ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-quandleorderG\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "linalg[coldim](convert(Q9,ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 3 " " 0 "" {TEXT -1 30 "III. Coloring braids with C" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 683 " This is the most exciting upgrade to t his project. Since the complexity of coloring braids with quandles is \+ O(c|X|^n) where |X| is the order of the quandle and n is the braid ind ex, the time required to quickly calculate the values of the invariant become limited very fast. For example, |X|=8 presents problems and wh en |X|=25 knots with a braid index of 3 are almost impossible to calcu late. This package will use a c program that was compiled as a shared \+ object file, ValCol8_1_05.so (dynamic link library in other O.S.'s), a nd then linked to Maple with the define_external procedure, validColor Vectors(). This greatly increased the size of the quandles we can work with." }}{PARA 0 "" 0 "" {TEXT 200 136 " This program will on ly color the braids of the knot. Some issues regarding the coloring of the regions will be discussed below." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 368 " The rest of this section explains how this is \+ accomplished. Knowledge of this material does not limit ones ability t o use the other procedures. This is all hidden in the functioning of t he procedures that calculate the values of the invariants. This will h owever be necessary if one wishes to create his or her own procedures \+ that use this C coloring program." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 155 " The C source code for the braid coloring program is \+ printed below with a system call.\nNote: This is Linux specific since \+ the cat function is used." }}{PARA 0 "" 0 "" {TEXT -1 3121 " Th is program will generate all possible initial color vectors and then c olor a given braid. Only the valid color vectors are saved. These colo r vectors are then stored in an array (pointer) validCvecs that is pas sed to the c program from this Maple procedure. The parameters used in the define_external are now explained. The fist element ''validcolorv ectorsint' is the name of the function that was used in the c program. 'quandle' is the Cayley table for the quandle that is used. Arrays mu st be modified slightly before passing them to the c program. Assume y ou have a quandle, Quandle:=Q3 := array(0 .. 2, 0 .. 2,[(2, 1)=0,(0, 0 )=0,(2, 2)=2,(0, 1)=2,(0, 2)=1,(1, 0)=2,(1, 1)=1,(1, 2)=0,(2, 0)=1]); \+ stored as in the previous worksheets. Now the quandle should be modifi ed to QuandleC by the following statement, QuandleC:=Array(op(Quandle) ,datatype=integer[2],order=C_order): This will change the datatype to \+ a 2 byte integer and store it in C_order (one dimensional zero indexed array). This will allow the c program to use it. All arrays passed to this function must be modified in this way. The second term 'quandles ize' is an integer representing the order of the quandle. 'knot' is an array that is the braid word for the knot. It must be modified as the quandle array was before passing it. Next the braid index and the num ber of crossings, 'braidindex' and 'nopsknot' respectively, are intege rs. Finally, an empty array validCvecs' is created and passed to the f unction. This is really just a pointer that will be used to store that valid color vectors. This array can be initialized by the following s tatement, VALIDCOLORS:=Array(0..10000000,datatype=integer[2],order=C_o rder):\nThis array is declared with a bound to it (10000000 in this ex ample). Statically declaring the array makes things easier, but may cr eate problems under special cases. For example, it requires (braidind ex * nops(Knot)+1) many cells of the array to store the valid color ve ctors for one valid coloring of the braid. So if the # valid colorings is greater than 10000000/(braidindex*nops(Knot)+1) then a segmentatio n fault will be created with the c program and then passed to the Mapl e program. This is not a good programming practice, but it will work f or now given our other constraints. There are some possible modificati ons that can be made without rewriting the c program. One such solutio n would be to allocate more memory for the validCvecs array. The above example allotted 10000000 cells at 2 bytes each. This requires 20 MB \+ of space. So if your computer has enough memory you could simply decla re a larger array. There are other solutions to this problem. If this \+ turns out to be a frequently encountered problem, we will rework this. I have ideas already, but I feel this is a good alpha version. The fu nction will return an integer representing the number of valid colorin gs for the braid. This is a 4 byte integer and should not exceed 2^31 \+ -1. The shared object file \"ValCol8_1_05.so\" should be stored in the same directory as the worksheet calling it to avoid altering the PATH given to the LIB statement below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "#alter path here to location of the c file.\n\nDD:=s system(\"cat /media/sda1/WorksheetIV/libcolor8_1_05.c\"):DD[2];" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#Qf^x//~A~c~program~to~cal~color~vector s~to~run~in~maple.~Trying~to~reslove~a~conflict|+//7/22/05|+//return~n umbcolors,~typecase~to~short|+//8_1_05~removed~some~unnecessary~variab les.~Changed~some~declaration~to|+//~short~int.~removed~typecase~to~va lidCvecs~assignment.~Removed~validCvecs[0].|+//~~Indexed~validCvecs~to ~1~not~brind*(nopsknot+1).~|+|+#include~|+#include~|+ |+|+int~eqarray(short~int~*A,short~int~*B,~int~nops);|+|+|+int~validco lorvectorsint(short~*quandle,|+~~~~~~~~~~~~~~~~~~~~~~~~~int~quandlesiz e,|+~~~~~~~~~~~~~~~~~~~~~~~~~short~*knot,|+~~~~~~~~~~~~~~~~~~~~~~~~~in t~braidindex,|+~~~~~~~~~~~~~~~~~~~~~~~~~int~nopsknot,|+~~~~~~~~~~~~~~~ ~~~~~~~~~~short~*validCvecs)|fr|+|+|+|+|+|+~~short~quandleinv[quandles ize*quandlesize];|+~~int~temp;~~|+|+~~int~tempscript=0;|+|+~~int~numbc olors=0;|+|+|+|+|+//~~~int~count=1;|+~~int~k;|+~~int~contcoloring=0;|+ |+|+~~short~int~color[nopsknot+2][braidindex];~~//~goes~1~to~brind|+~~ |+~~short~int~carryover[braidindex+2];|+~~short~int~temparray[braidind ex+2];|+|+~~int~colordiff=0;|+~~int~i,j,h,jj,kk;|+//~int~helpme,ht;|+~ ~int~nCo=0;|+~~|+|+~~~for(i=0;i~0)|fr|+~~ ~~~~for(h=1;~h<=braidindex;h++)|fr|+|*~~if(~h==abs(knot[j-1]))|+~~~~~~ ~~~~~~~~~~~color[j+1][h]=color[j][h+1];|+|*~~~~~~~~~~~~|+|*~~if(~h==(a bs(knot[j-1])+1))|+|*~~~~~~color[j+1][h]=quandle[color[j][h-1]*quandle size+color[j][h]];|+|*~~~~~~~~~~~~~~|+|*~~if(~h(abs(kno t[j-1])+1))|+~~~~~~~~~~~~~~~~~~~color[j+1][h]=color[j][h];|+|*~~~~~~|+ ~~~~~~~|hr|+~~~~|hr|+~~~~else|fr|+~~~~~~for(h=1;~h<=braidindex;h++)|fr |+|*~~if(~h==abs(knot[j-1]))|+~~~~~~~~~~~~~~~color[j+1][h]=quandleinv[ color[j][h]*quandlesize+color[j][h+1]];|+|*~~if(~h==(abs(knot[j-1])+1) )|+~~~~~~~~~~~~~~color[j+1][h]=color[j][h-1];|+|*~~if(~h(abs(knot[j-1 ])+1))|+~~~~~~~~~~~~~~color[j+1][h]=color[j][h];|+~~~~~~~|hr|+~~~~~|hr |+~~|hr|+~colordiff=0;|+|+~~colordiff=eqarray(color[1],color[nopsknot+ 1],braidindex);|+~~if(colordiff)|fr|+|+//~~~~~~validCvecs[0]++;|+//~~~ ~printf(\"\\n\\n\");|+~~~~for(jj=1;jj<=nopsknot+1;jj++)|fr|+~~~~~~~for (kk=1;kk<=braidindex;kk++)|fr|+//~~~~~~~~~printf(\"%d~~~~~\",color[jj] [kk]);|+~~~~~~~tempscript=numbcolors*(nopsknot+1)*braidindex+((jj-1)*b raidindex+kk);|+~~~~~~~validCvecs[tempscript]=~color[jj][kk];|+~~~~~~~ |hr|+//~~~~~~~~~printf(\"\\n\\n\");|+~~~~|hr|+~~~~~~numbcolors++;|+~~~ ~~~nCo++;|+|hr~~|+~~|+|+|+contcoloring=eqarray(~color[1],~temparray,br aidindex);|+~|+|+~~~color[1][1]=color[1][1]+1;|+~~~~k=1;|+~~~~carryove r[k]=~color[1][k]/quandlesize;|+|+|+~~~~while(carryover[k]!=0&&k " 0 "" {MPLTEXT 1 0 58 "QuandleC:=Array(op(R3),datat ype=integer[2],order=C_order);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Q uandleCG-%'RTABLEG6%\"*7A_P\"-%&ArrayG6(;\"\"!\"\"#F,<(/6$F.F.F./6$F- \"\"\"F./6$F-F.F4/6$F4F-F./6$F4F4F4/6$F.F-F4/%)datatypeG&%(integerG6#F ./%(storageG%,rectangularG/%&orderG%(C_orderGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 50 "Next convert the knot to the appropriate datatype. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "KnotC:=Array(0..(nops(K not[0,2])-1),Knot[0,2],datatype=integer[2],order=C_order);\n\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&KnotCG-%'RTABLEG6%\"*S;!z8-%&ArrayG 6';\"\"!\"\"#<%/6#F-\"\"\"/6#F2F2/6#F.F2/%)datatypeG&%(integerGF6/%(st orageG%,rectangularG/%&orderG%(C_orderGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 188 "Now we need a place to store the valid color vectors th at the C program will find (a pointer). Notice how the Memory tab in t he bottom right of the window will jump after this declaration." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "VALIDCOLORS:=Array(0..100000 00,datatype=integer[2],order=C_order);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,VALIDCOLORSG-%'RTABLEG6*\"*S*zZ8&%(integerG6#\"\"#%&ArrayG%,r ectangularG%(C_orderG7\"\"\"\";\"\"!\")+++5" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 200 76 "Now for the magic! We call the program to calculate th e valid color vectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 " Knot[0,2];\nquandleorder:=linalg[coldim](convert(R3,matrix));\nbrind:= max(op(map(x->abs(x),Knot[0,2])))+1;\nnumcross:=nops(Knot[0,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-quandleorderG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&brindG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)numcrossG\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "numVcolors:=validColorV ectors(QuandleC,quandleorder,KnotC,brind,numcross,VALIDCOLORS):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 110 "The number of valid colorings is the return type. In this example we assigned this to the variable num Vcolors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "numVcolors;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 256 "So the trefoil has 9 valid colorings when colored with \+ the 3 element Dihedral quandle. Recall that the number of colorings of a knot with a quandle is an invariant in itself. This procedure could be used by itself to calculate the values of that invariant." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 208 "The array that is holding the va lid color vectors was passed to the procedure. We allocated to much sp ace for this (intentionally). The value stored in the array will not \+ print since the array is so large. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "VALIDCOLORS;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RT ABLEG6*\"*S*zZ8&%(integerG6#\"\"#%&ArrayG%,rectangularG%(C_orderG7\"\" \"\";\"\"!\")+++5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 177 "There were only 9 valid colorings in this case, so there should be data in 72 ce lls of the array. Any cells beyond this point are zero. The value that they were initialized to. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for i from 0 to 10*4*2 do\n printf(\"%d\",VALIDCOLORS[i]);\nod;" }}{PARA 6 "" 1 "" {TEXT -1 81 "000000000100221102001122001122001111111 112110022102211002122001122222222200000000" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 550 "So we have the valid colorings, but they are stored in \+ an undesirable way and we using a lot of memory that is not necessary. We will not use the procedure cvrtarray() to manipulate the data and \+ store it in a better way. The procedure cvrtarray:=proc(brind,nopknt,L ,numVC) must be passed the braid index, number of crossing, the VALIDC OLORS obtained from the validColorVectors() call, and finally the numb er of valid colorings (the value returned from the procedure validColo rVectors()). We will assign the valid coloring to the array called Co lors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Colors:=cvrtarray( 2,3,VALIDCOLORS,numVcolors);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Col orsG-%&TABLEG6#7,/\"\"!\"\"*/\"\"\"-F&6#7&/F--%'vectorG6#7$F*F*/\"\"#F 2/\"\"$F2/\"\"%F2/F7-F&6#7&/F--F36#7$F-F*/F7-F36#7$F*F7/F9-F36#7$F7F-/ F;FA/F9-F&6#7&/F--F36#7$F7F*/F7-F36#7$F*F-/F9-F36#7$F-F7/F;FR/F;-F&6#7 &/F-FV/F7FZ/F9FR/F;FV/\"\"&-F&6#7&/F--F36#7$F-F-/F7Ffo/F9Ffo/F;Ffo/\" \"'-F&6#7&/F-FI/F7FA/F9FE/F;FI/\"\"(-F&6#7&/F-FE/F7FI/F9FA/F;FE/\"\")- F&6#7&/F-FZ/F7FR/F9FV/F;FZ/F+-F&6#7&/F--F36#7$F7F7/F7F\\r/F9F\\r/F;F\\ r" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 289 "The array Colors is a thre e dimensional array, Colors[i][j][k]. The first index, i, ranges from \+ 1 to the number of valid colorings. The second and third are the posit ion of the braid where that color was assigned. The procedure printCol ors:=proc(Knot,Col,numVc) will print these colorings." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "printColors(Knot[0,2],Colors,numVco lors);" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011\011 1 0 \011\011 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0 \011\011 0 2\011\011 0 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011\011 2 1\011\011 1 2\011\011" }} {PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011\011 1 0\011\011 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 1\011\011 1 \+ 1\011\011 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 1 \+ 2\011\011 1 1\011\011 1 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 2 0\011\011 1 1\011\011 0 2\011\011" } }{PARA 6 "" 1 "" {TEXT -1 36 " 0 1\011\011 1 1\011\011 \+ 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 2\011\011 1 \+ 2\011\011 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 2 \+ 1\011\011 2 0\011\011 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 1 0\011\011 0 1\011\011 2 2\011\011" } }{PARA 6 "" 1 "" {TEXT -1 36 " 0 2\011\011 1 2\011\011 \+ 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 77 "Now lets examine the time it take s to color when we use much larger quandles." }}{PARA 0 "" 0 "" {TEXT 200 323 "This first example will color the figure eight knot with a 25 element quandle, Z_5[t,t^(-1)]/(t^2-t+1). We will use two system cal ls to the \"date\" function to examine the time required for the c pro gram to check which of the 15625 possible initial color vectors produc e valid colorings. This system call is Linux specific." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "Q25:=AlexQuandle(t^2-t+1,5):\nQuan dleC:=Array(op(Q25),datatype=integer[2],order=C_order):\nKnotC:=Array( 0..(nops(Knot[1,2])-1),Knot[1,2],datatype=integer[2],order=C_order):\n VALIDCOLORS:=Array(0..10000000,datatype=integer[2],order=C_order):\nKn ot[1,1];Knot[1,2];\nquandleorder:=linalg[coldim](convert(Q25,matrix)): \nbrind:=max(op(map(x->abs(x),Knot[1,2])))+1:\nnumcross:=nops(Knot[1,2 ]):\nHH:=ssystem(date):HH[2];\nnumVcolors:=validColorVectors(QuandleC, quandleorder,KnotC,brind,numcross,VALIDCOLORS):\nHH:=ssystem(date):HH[ 2];\nnumVcolors;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q$4_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"!\"#F$F%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q>Tue~Oct~11~13:59:43~EDT~2005|+6\"" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#Q>Tue~Oct~11~13:59:43~EDT~2005|+6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 81 "This o nly had the trivial colorings. We can see them with the following comm ands." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Colors:=cvrtarray( 3,4,VALIDCOLORS,numVcolors):\nprintColors(Knot[1,2],Colors,numVcolors) ;" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011\011 1 1 \+ 1\011\011 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 0 0 0\011\011 1 1 1\011\011 2 2 2\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011\011 1 1 1 \011\011 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 \+ 0 0\011\011 1 1 1\011\011 2 2 2\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011\011 1 1 1 \011\011 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 \+ 3 3\011\011 4 4 4\011\011 5 5 5\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 3 3\011\011 4 4 4\011\011 \+ 5 5 5\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 3 3 \011\011 4 4 4\011\011 5 5 5\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 3 3\011\011 4 4 4\011\011 5 \+ 5 5\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 3 3\011 \011 4 4 4\011\011 5 5 5\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 6 6 6\011\011 7 7 7\011\011 8 8 8 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 6 6 6\011\011 7 \+ 7 7\011\011 8 8 8\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 6 6 6\011\011 7 7 7\011\011 8 8 8\011 \011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 6 6 6\011\011 7 7 7\011\011 8 8 8\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 6 6 6\011\011 7 7 7\011\011 8 8 8\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 51 " 9 9 9\011\011 10 10 10 \011\011 11 11 11\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 9 \+ 9 9\011\011 10 10 10\011\011 11 11 11\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 9 9 9\011\011 10 10 10 \011\011 11 11 11\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 9 \+ 9 9\011\011 10 10 10\011\011 11 11 11\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 9 9 9\011\011 10 10 10 \011\011 11 11 11\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 12 1 2 12\011\011 13 13 13\011\011 14 14 14\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 12 12 12\011\011 13 13 13\011\011 \+ 14 14 14\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 12 12 12 \011\011 13 13 13\011\011 14 14 14\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 12 12 12\011\011 13 13 13\011\011 14 \+ 14 14\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 12 12 12\011 \011 13 13 13\011\011 14 14 14\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 15 15 15\011\011 16 16 16\011\011 17 17 17 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 15 15 15\011\011 16 \+ 16 16\011\011 17 17 17\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 15 15 15\011\011 16 16 16\011\011 17 17 17\011 \011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 15 15 15\011\011 16 16 16\011\011 17 17 17\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 15 15 15\011\011 16 16 16\011\011 17 17 17\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 51 " 18 18 18\011\011 19 19 19 \011\011 20 20 20\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 18 \+ 18 18\011\011 19 19 19\011\011 20 20 20\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 18 18 18\011\011 19 19 19 \011\011 20 20 20\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 18 \+ 18 18\011\011 19 19 19\011\011 20 20 20\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 18 18 18\011\011 19 19 19 \011\011 20 20 20\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 21 2 1 21\011\011 22 22 22\011\011 23 23 23\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 21 21 21\011\011 22 22 22\011\011 \+ 23 23 23\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 21 21 21 \011\011 22 22 22\011\011 23 23 23\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 21 21 21\011\011 22 22 22\011\011 23 \+ 23 23\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 21 21 21\011 \011 22 22 22\011\011 23 23 23\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 17 " 24 24 24\011\011" }}{PARA 6 "" 1 "" {TEXT -1 17 " 24 \+ 24 24\011\011" }}{PARA 6 "" 1 "" {TEXT -1 17 " 24 24 24\011 \011" }}{PARA 6 "" 1 "" {TEXT -1 17 " 24 24 24\011\011" }}{PARA 6 "" 1 "" {TEXT -1 17 " 24 24 24\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 200 126 "It took less than one second to check all of the poss ible colorings, and it found that only 25 ( the trivial ones) were val id." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 198 "That was too fast!! Sinc e the complexity is O(c|X|^braidindex), lets increase the size of the \+ quandle to 256 elements and use the same knot (braid index of 3).\nThi s example should not take to long. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 530 "Q256:=AlexQuandle(t^8+1,2):\nQuandleC:=Array(op(Q256 ),datatype=integer[2],order=C_order):\nKnotC:=Array(0..(nops(Knot[1,2] )-1),Knot[1,2],datatype=integer[2],order=C_order):\nVALIDCOLORS:=Array (0..10000000,datatype=integer[2],order=C_order):\nKnot[1,1];Knot[1,2]; \nquandleorder:=linalg[coldim](convert(Q256,matrix)):\nbrind:=max(op(m ap(x->abs(x),Knot[1,2])))+1:\nnumcross:=nops(Knot[1,2]):\nHH:=ssystem( date):HH[2];\nnumVcolors:=validColorVectors(QuandleC,quandleorder,Knot C,brind,numcross,VALIDCOLORS):\nHH:=ssystem(date):HH[2];\nnumVcolors; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q$4_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"!\"#F$F%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q>T ue~Oct~11~14:00:36~EDT~2005|+6\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q> Tue~Oct~11~14:00:47~EDT~2005|+6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"$c#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 713 "The time stamps shows \+ that it takes about 12 seconds (on my computer also this time will var y) to calculate the valid colorings. FAST!! FAST!! When you run this \+ it did take longer than that however. This is mostly due to the cost o f generating the Cayley table for the quandle. Increasing the braid in dex will obviously slow this program down much faster than increasing \+ the size of the quandle. This example only had to generate and check 1 6777216 many colorings. Using the 25 element quandle above with knot \+ \"10_35\" would require generating and checking 244140625 possible col orings. This took about 13 minutes (on my computer). You can uncomment out the below execution group if you wish to run this example. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "#Q25:=AlexQuandle(t^2-t+1,5 ):\n#QuandleC:=Array(op(Q25),datatype=integer[2],order=C_order):\n#Kno tC:=Array(0..(nops(Knot[118,2])-1),Knot[118,2],datatype=integer[2],ord er=C_order):\n#VALIDCOLORS:=Array(0..10000000,datatype=integer[2],orde r=C_order):\n#Knot[118,1];Knot[118,2];\n#quandleorder:=linalg[coldim]( convert(Q25,matrix)):\n#brind:=max(op(map(x->abs(x),Knot[118,2])))+1: \n#numcross:=nops(Knot[118,2]):\n#HH:=ssystem(date):HH[2];\n#numVcolor s:=validColorVectors(QuandleC,quandleorder,KnotC,brind,numcross,VALIDC OLORS):\n#HH:=ssystem(date):HH[2];\n#numVcolors;" }}}{EXCHG {PARA 3 " " 0 "" {TEXT -1 44 " IV. Calculating the value of the invariant" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 " This section covers the p rocedures:" }}{PARA 0 "" 0 "" {TEXT -1 127 "Invar2_3UntwistedDLL:=proc (Quandle,p,Knot);\nInvar2UntwistedDLL:=proc(Quandle,p,Knot);\nInvar3Un twistedDLL:=proc(Quandle,p,Knot);" }}{PARA 0 "" 0 "" {TEXT -1 118 "Inv ar2_3MochizukiDLL:=proc(Xpolym,Apolym,p,m1,m2,a3,n1,b2,Knot);\nInvar2M ochizukiDLL:=proc(Xpolym,Apolym,p,n1,b2,Knot);" }}{PARA 0 "" 0 "" {TEXT -1 1021 "Invar3MochizukiDLL:=proc(Xpolym,Apolym,p,m1,m2,a3,Knot) ;\nInvar2TwistedDLL(Quandle,knot,polym,p);.\nEach of these procedures \+ will calculate the value of the invariant for a specific knot for the \+ given cocycle case. The coloring is performed with the c program and h idden from the user. These programs are otherwise similar to that in t he previous worksheets. When it made sense, procedures calculating the 2-cocycle and 3-cocycle cases at the same time were created. The colo rings of the regions is done with Maple. This was intentional since we know that there are |X|*m, where |X| is the order of the quandle and \+ m is the number of valid colorings of the braid, validcolorings when s hadow coloring. Since the number of valid colorings of the braid is u sually small, relatively speaking, the cost for coloring the regions s hould be small. Hence the decision to use Maple. There are examples wh ere this approach has created trouble, high time cost. Coloring of the regions with c will help and will be implemented soon. " }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 14 "Untwiste d case" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 841 " The procedure Invar2_3UntwistedDLL:=proc(Quandle,p,Knot) will solve the 2-cocycle a nd 3-cocycle conditions and calculate the values of each of the invari ants. The procedure requires three parameters. The first is the quandl e passed as before, a two dimensional zero indexed array representing \+ the Cayley table of the quandle. The second is the prime modulus for t he Coefficient group, Z_p. Finally the knot in braid form. This proced ure will return 6 items. The first and second are the values of the un twisted 2-cocycle invariant and untwisted 3-cocycle invariant, respect ively. The third is the valid coloring of the braid. Fourth and fifth \+ are the solutions to the 2 and 3-cocycle conditions. Finally, the numb er of valid coloring of the braid is returned. By assigning these valu es to a list we can access only the desired data. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 160 " This example will calculate value of the untwisted 2 and 3-cocycle invariant for the figure eight knot with th e quandle Q4_3 and coefficient group Z_2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "print(convert(Q4_3,Matrix));\nKnot[1,1];Knot[1,2]; \nItems:=Invar2_3UntwistedDLL(Q4_3,2,Knot[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"+3Z#f4\"-%'MATRIXG6#7&7&\"\"!\"\"$\"\"\" \"\"#7&F/F.F-F,7&F-F,F/F.7&F.F/F,F-%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q$4_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"!\" #F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/co3solproccallG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/co2solproccallG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ItemsG6(,&\"\"%\"\"\"*&\"#7F()%\"uG,(&%\"tG6#\"#9F(& F/6#\"#5F(&F/6#\"#:F(F(F(,&\"#;F(*&\"#[F()F,,*&F/6#\"#iF(&F/6#\"#cF(&F /6#\"#`F(&F/6#\"#ZF(F(F(%&ColorG%\"fG%\"hGF9" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 200 74 "First we get the value of the untwisted 2-cocycle inva riant from Items[1]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Item s[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"%\"\"\"*&\"#7F%)%\"uG,( &%\"tG6#\"#9F%&F,6#\"#5F%&F,6#\"#:F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 68 "We get the value of the untwisted 3-cocycle invariant fr om Items[2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Items[2];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#;\"\"\"*&\"#[F%)%\"uG,*&%\"tG6# \"#iF%&F,6#\"#cF%&F,6#\"#`F%&F,6#\"#ZF%F%F%" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 200 310 "Note: The t[i] in the values of the invariant are fre e variables obtained from solving the cocycle conditions. They take va lues from the Z_p. By assigning values to these free variables we are \+ in essence choosing a specific cocycle and getting a specific value. T his output is a family of values of invariants." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 225 "If one chooses the valid coloring of the braid can then be saved or viewed. We will use the print Colors() procedure to \+ view the 16 different colorings of the braid here. The number of valid coloring of the braid is Items[6]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "printColors(Knot[1,2],Items[3],Items[6]);" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011\011 3 1 0\011\011 \+ 1 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0 \011\011 1 2 0\011\011 2 3 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011\011 1 3 2\011\011 2 \+ 1 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011 \011 3 0 2\011\011 1 0 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 0 0\011\011 3 1 0\011\011 1 2 \+ 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 2 3 0\011\011 2 \+ 0 1\011\011 1 1 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 1 0\011\011 0 3 1\011\011 1 1 1\011\011 " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 2 1\011\011 0 2 \+ 3\011\011 1 1 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 2 0 1\011\011 2 1 3\011\011 1 1 1\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 2 3 0\011\011 2 0 1 \011\011 1 1 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 \+ 2 1\011\011 0 3 1\011\011 3 0 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 2 0 1\011\011 3 2 1\011\011 \+ 0 1 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 2 3 0 \011\011 3 0 2\011\011 0 3 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 1 0\011\011 0 1 2\011\011 3 \+ 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 2 1\011 \011 0 3 1\011\011 3 0 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 1 2\011\011 2 2 2\011\011 1 3 2 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 1 3 2\011\011 2 \+ 2 2\011\011 3 0 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 1 0 3\011\011 2 2 2\011\011 3 1 0\011 \011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 2 3\011\011 2 2 2\011\011 1 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 0 1 2\011\011 2 2 2\011\011 1 3 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 51 " 1 0 3\011\011 2 1 3 \011\011 0 2 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 0 \+ 2 3\011\011 1 0 3\011\011 2 1 3\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 0 1 2\011\011 1 2 0 \011\011 2 0 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 51 " 1 \+ 3 2\011\011 2 3 0\011\011 0 3 1\011\011" }} {PARA 6 "" 1 "" {TEXT -1 51 " 1 0 3\011\011 2 1 3 \011\011 0 2 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 17 " 3 \+ 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 17 " 3 3 3\011\011 " }}{PARA 6 "" 1 "" {TEXT -1 17 " 3 3 3\011\011" }}{PARA 6 " " 1 "" {TEXT -1 17 " 3 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 17 " 3 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 63 " The solutions to the untwisted 2-cocycle condition is Items[4]." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "print(Items[4]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&ARRAYG6$7$;\"\"!\"\"$F'72/6$F(F(F(/6$F(\" \"\",&&%\"tG6#\"#9F/&F26#\"#5F//6$F(\"\"#,(F1F/F5F/&F26#\"#8F//6$F(F), &&F26#\"#:F/F1F//6$F/F(,(F5F/FBF/F " 0 "" {MPLTEXT 1 0 16 "print(Items[5]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#-%&ARRAYG6$7%;\"\"!\"\"$F'F'7\\o/6%F(F (F(F(/6%F(F(\"\"\",&&%\"tG6#\"#iF/&F26#\"#UF//6%F(F(\"\"#,(F1F/F5F/&F2 6#\"#hF//6%F(F(F),&&F26#\"#jF/F1F//6%F(F/F(,*&F26#\"#eF/&F26#\"#`F/&F2 6#\"#JF/&F26#\"#dF//6%F(F/F/F(/6%F(F/F:,4FBF/F1F/&F26#\"#gF/FHF/&F26# \"#bF/&F26#\"#cF/&F26#\"#ZF/F5F/FNF//6%F(F/F),2FHF/FfnF/FKF/F5F/F1F/FB F/F\\oF/FinF//6%F(F:F(,6F1F/FYF/F " 0 "" {MPLTEXT 1 0 92 "print(convert(Q5_4 ,matrix));\nKnot[0,1];\nKnot[0,2];\nL:=Invar3UntwistedDLL(Q5_4,2,Knot[ 0,2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"!F(F(F (F(7'\"\"\"F*\"\"%\"\"#\"\"$7'F,F-F,F+F*7'F-F+F*F-F,7'F+F,F-F*F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q$3_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/co3solproccall G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG6&,(\"#D\"\"\"*&\"#7F()%\"u G,(&%\"tG6#\"#=F(&F/6#\"#CF(&F/6#\"#BF(F(F(*&\"#[F()F,,(&F/6#\"$B\"F(& F/6#\"$?\"F(&F/6#\"$9\"F(F(F(%&ColorG%\"hG\"#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 47 "The value of the untwisted 3-cocycle invariant." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"#D\"\"\"*&\"#7F%)%\"uG,(&%\"tG6#\"#=F%&F,6#\"#CF%&F ,6#\"#BF%F%F%*&\"#[F%)F),(&F,6#\"$B\"F%&F,6#\"$?\"F%&F,6#\"$9\"F%F%F% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 224 "Recall that this procedure \+ only returns the colorings of the braids, L[2], and the number of vali d colorings of the braid, L[4]. Not the colors and number of colorings of the braid and regions. The 2-cocycle case is similar." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 12 "Twisted Case" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 600 " There is only one procedure for this case, th e twisted 2-cocycle. The form that the value of the invariant is retur ned in has changed some from the version that appeared in worksheet II . In worksheet II the value was returned as a multiset where the eleme nts were the state-sum contribution of a specific coloring. The value \+ that is returned now will appear more like the untwisted case. This pr ocedure requires the following: Quandle, Knot, polynomial in t for the coefficient group, and prime modulus for the coefficient group. The r eturn type is similar to the untwisted 2-cocycle case. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 117 " Let's look at an example. We will use \+ the quandle Q3 for the knot \"6_1\" and coefficient group Z_3[t,t^(-1) ]/(t^2+1)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "print(convert (Q3,matrix));\nKnot[4,1];\nKnot[4,2];\nK:=Invar2TwistedDLL(Q3,Knot[4,2 ],t^2+1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!\" \"#\"\"\"7%F)F*F(7%F*F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q$6_16\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"\"F$\"\"#!\"\"!\"$F%F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG6&,&\"\"$\"\"\"*&\"\"'F()%\"uG,( *&,(&%\"xG6#F(F(&F16#\"\"#F(*&F5F(&F16#F'F(F(F(%\"tGF(F(F7F(F0F(F(F(%& ColorG%\"FG\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 127 "The value o f the twisted 2-cocycle invariant for the quandle Q3 for the knot \"6_ 1\" and coefficient group Z_3[t,t^(-1)]/(t^2+1)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "K[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\" \"$\"\"\"*&\"\"'F%)%\"uG,(*&,(&%\"xG6#F%F%&F.6#\"\"#F%*&F2F%&F.6#F$F%F %F%%\"tGF%F%F4F%F-F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 299 "The x[i] are free variable introduced when solving the twisted 2-cocycle \+ conditions. These free variable are carried over to the value of the i nvariant. t is the indeterminate from the coefficient group. By assign ing values to these free variables we get a specific element of the co efficient group." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 38 "The nine val id colorings of the braid." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "printColors(Knot[4,2],K[2],K[4]);" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0\011\011 0 1 1 0\011\011 0 2 \+ 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0 \011\011 1 2 1 0\011\011 2 1 2 0\011\011" }} {PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0\011\011 2 0 \+ 1 0\011\011 1 0 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0\011\011 2 1 2 0\011\011 1 \+ 2 1 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 \+ 0\011\011 0 2 2 0\011\011 0 1 1 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0\011\011 0 2 \+ 1 2\011\011 0 1 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0\011\011 0 1 0 2\011\011 \+ 0 2 0 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 0 0 0\011\011 0 1 1 0\011\011 0 2 2 0\011 \011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 66 " 1 0 0 1\011\011 1 1 \+ 1 1\011\011 1 2 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 2 0 1\011\011 1 1 1 1\011\011 \+ 2 0 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 2 1 0 1\011\011 1 1 1 1\011\011 0 1 2 1\011 \011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 2 0 2 1\011\011 1 1 1 1\011\011 0 2 0 1\011\011" }}{PARA 6 "" 1 " " {TEXT -1 66 " 1 2 2 1\011\011 1 1 1 1\011 \011 1 0 0 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 1 2 0 2\011\011 1 1 1 1\011\011 1 0 2 \+ 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 1 0 1 2\011\011 1 1 1 1\011\011 1 2 1 0\011\011" }}{PARA 6 " " 1 "" {TEXT -1 66 " 1 0 0 1\011\011 1 1 1 1 \011\011 1 2 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 66 " 2 \+ 0 0 2\011\011 2 1 1 2\011\011 2 2 2 2 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 0 1 0 2\011\011 \+ 1 0 1 2\011\011 2 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 1 2 0 2\011\011 0 2 1 2\011 \011 2 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 1 0 1 2\011\011 0 1 0 2\011\011 2 2 2 \+ 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 2 1 1 2\011\011 2 0 0 2\011\011 2 2 2 2\011\011" }}{PARA 6 " " 1 "" {TEXT -1 66 " 2 1 0 1\011\011 2 0 1 0 \011\011 2 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " \+ 2 0 2 1\011\011 2 1 2 0\011\011 2 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 66 " 2 0 0 2\011 \011 2 1 1 2\011\011 2 2 2 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 50 " The solutions to the twisted 2-cocycle conditions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "print(K[3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7+/6$\"\"!\"\"#,(*&,&&%\"xG6#\"\"\"F1*&F*F1&F/6#F*F1F 1F1%\"tGF1F1*&F*F1&F/6#\"\"%F1F1*&F3F1F5F1F1/6$F*F1,(*&,(F.F1F3F1*&F*F 1&F/6#\"\"$F1F1F1F5F1F1*&F*F1F7F1F1*&F.F1F5F1F1/6$F*F*F)/6$F)F1,(*&,&F 3F1FAF1F1F5F1F1F7F1*&,(*&F*F1F.F1F1F3F1FAF1F1F5F1F1/6$F*F),(*&,&*&F*F1 F3F1F1*&F*F1FAF1F1F1F5F1F1F7F1*&,(F.F1*&F*F1F3F1F1FAF1F1F5F1F1/6$F)F)F )/6$F1F1F)/6$F1F*,&F7F1*&FAF1F5F1F1/6$F1F),(*&,&FAF1F.F1F1F5F1F1*&F*F1 F7F1F1*&,&*&F*F1F.F1F1*&F*F1F3F1F1F1F5F1F1" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 33 "Mochizuki polynomial cocycle case" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 117 " In this case, both the 2 and 3-cocycles \+ make sense. So we have three procedures to calculate these values, " } {TEXT -1 195 "Invar2_3MochizukiDLL:=proc(Xpolym,Apolym,p,m1,m2,a3,n1,b 2,Knot), Invar2MochizukiDLL:=proc(Xpolym,Apolym,p,n1,b2,Knot), and Inv ar3MochizukiDLL:=proc(Xpolym,Apolym,p,m1,m2,a3,Knot). The procedure, \+ " }{TEXT 200 1 " " }{TEXT -1 762 "Invar2_3MochizukiDLL(), for calculat ing the 2-cocycle and 3-cocycle case at the same time will be describe d. The other procedures are similar for the individual cases. Xpolym i s a polynomial in t used for the Alexander quandle used for coloring t he braids. Apolym is the polynomial in t used for the coefficient grou p. p is the prime modulus used for both the Alexander quandle and coef ficient group. m1, m2, and a3 correspond to the exponents from Mochizu ki's 3-cocycle formula, f(x,y,z)=(x-y)^(p^m1) * (y-z)^(p^m2) * z^(a3). This procedure will check that the Mochizuki condition are satisfied. ie. Apolym divides 1-t^((p^m1)+(p^m2)+a_3) mod p where a3 is a power of p or that a3=0. n1 and b2 correspond to the exponents for the Moch izuki 2-cocycle formula, " }}{PARA 0 "" 0 "" {TEXT -1 156 "f(x,y)=(x- y)^(p^n1) * y^(b2). The Mochizuki 2-cocycle conditions are also checke d in this case. The six items returned are ordered like the untwisted \+ case. " }}{PARA 0 "" 0 "" {TEXT 200 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 318 "This example will calculate the values of the invariant s for the trefoil knot where the quandle and coefficient group are Z_2 [t,t^(-1)]/(t^2-t+1) The 2-cocycle formula is f(x,y)=(x-y)^2 * y so we assign the values n1=1, b2=1. The3-cocycle formula is f(x,y,z)=(x-y) ^ * (y-z)^2 so we assign the values m1=0, m2=1, a3=0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "print(convert(AlexQuandle(t^2-t+1, 2),matrix));\nKnot[0,1];\nKnot[0,2];\nL:=Invar2_3MochizukiDLL(t^2-t+1, t^2-t+1,2,0,1,0,1,1,Knot[0,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' matrixG6#7&7&\"\"!\"\"$\"\"\"\"\"#7&F+F*F)F(7&F)F(F+F*7&F*F+F(F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q$3_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG6(,&\"\"% \"\"\"*&\"#7F()%\"uG,&%\"tGF(F(F(F(F(,&\"#;F(*&\"#[F()F,F.F(F(%&ColorG %\"fG%\"fGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 186 "The value of th e 2-cocycle invariant with quandle and coefficient group Z_2[t,t^(-1)] /(t^2-t+1) for the trefoil knot and 2-cocycle formula f(x,y)=(x-y)^2 \+ * y is the first item returned." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"%\"\"\"*&\"#7F%) %\"uG,&%\"tGF%F%F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 141 "The v alue of the 3-cocycle invariant with quandle and coefficient group Z_2 [t,t^(-1)]/(t^2-t+1) for the trefoil knot and 3-cocycle formula " }} {PARA 0 "" 0 "" {TEXT 200 54 "f(x,y,z)=(x-y)^ * (y-z)^2 is the second \+ item returned." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#;\"\"\"*&\"#[F%)%\"uG%\"tGF%F%" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 449 "The exponents of u in the valu e of the invariant are elements of the coefficient group Z_2[t,t^(-1)] /(t^2-t+1). t is the indeterminate. It is not a free variable. You wil l not encounter any free variables in this case. So in the 2-cocycle c ase there were 4 coloring that contributed the zero in Z_2[t,t^(-1)]/( t^2-t+1) as their state-sum contribution and 12 coloring that contribu ted t+1 in Z_2[t,t^(-1)]/(t^2-t+1) as their state-sum contribution. \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 161 "Once again the colors are j ust the valid coloring of the braid. We will print these by using the \+ third element returned and the sixth, number of valid colorings." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "printColors(Knot[0,2],L[3],L [6]);" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011\011 1 0\011 \011 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011 \011 0 2\011\011 0 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011\011 2 1\011\011 3 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 0\011\011 1 0\011\011 2 0 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 " 3 0\011\011 0 1\011 \011 1 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 1\011 \011 1 3\011\011 1 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 1 3\011\011 3 0\011\011 1 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 3 0\011\011 0 1\011\011 1 1 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 " 2 1\011\011 3 1\011 \011 0 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 1 0\011 \011 1 2\011\011 2 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 2\011\011 2 3\011\011 1 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 2 1\011\011 3 1\011\011 0 2 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 " 1 2\011\011 2 2\011 \011 3 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 2 3\011 \011 2 2\011\011 2 0\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 3 1\011\011 2 2\011\011 0 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 1 2\011\011 2 2\011\011 3 2 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 3\011\011 1 3\011 \011 2 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 3 2\011 \011 3 0\011\011 3 1\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 2 0\011\011 0 1\011\011 1 2\011\011" }}{PARA 6 "" 1 "" {TEXT -1 36 " 0 3\011\011 1 3\011\011 2 3 \011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 12 " 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 12 " 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 12 " \+ 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 12 " 3 3\011\011" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 81 "The fourth and fifth items return ed are the values from the Mochizuki formula's. " }}}{EXCHG {PARA 4 " " 0 "" {TEXT -1 50 "Mochizuki polynomial cocycle for Dihedral quandles " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 516 " There is not a sp ecial procedure for this case since it can be calculated by nesting ot her procedures. We will use the untwisted 3-cocycle procedure, Invar3U ntwistedDLL:=proc(Quandle,p,Knot), for this case. This procedure had a n optional fourth argument that was the cocycles. So if we use the pro cedure, MochizukiDihedral3coc:=proc(p), to calculate the values from t he Mochizuki formula, f[i,j,k]:=((i-j)*((2*k^p-j^p)-(2*k-j)^p)/p) mod \+ p and pass these as the fourth parameter we can calculate this case. \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 86 "This example is with the qua ndle R7, seven element Dihedral quandle, and the knot 5_2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Knot[3,1];\nKnot[3,2];\nM:=Invar3U ntwistedDLL(AlexQuandle(t+1,7),7,Knot[3,2],MochizukiDihedral3coc(7)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q$5_26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"F$F$\"\"#!\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG6&,*\"#\\\"\"\"*&\"#)*F()%\"uG\"\"$F(F(*&F*F()F,\"\"&F(F(* &F*F()F,\"\"'F(F(%&ColorG%\"fGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 36 "The value of the 3-cocycle invariant" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "M[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"#\\\"\"\" *&\"#)*F%)%\"uG\"\"$F%F%*&F'F%)F)\"\"&F%F%*&F'F%)F)\"\"'F%F%" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 59 " V. Mass calculations with autom atic file name generation" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 691 " \+ This worksheet and procedures were originally written with Maple \+ 10. The keyword optional parameters were not supported with Maple 8. \+ in the list of procedures below, you can spot these by,\{cpath=false\} ,\{solpath:=false\}. I modified the code for the procedures so that th is worksheet will function with Maple 8. Now the cpath and solpath ar e required parameters. they do not have an initial value. More about \+ this later in this section.\n This section consider the foll owing procedures:\nCalcUntwistinvars:=proc(filePATH,p ,qord,qnum,numbe rofknots,\{cpath=false\},\{solpath:=false\});\nCalc2Untwistinvars:=pro c(filePATH,p,qord,qnum,numberofknots,\{cpath=false\},\{solpath:=false \});" }}{PARA 0 "" 0 "" {TEXT -1 192 "Calc3Untwistinvars:=proc(filePAT H,p,qord,qnum,numberofknots,\{cpath=false\},\{solpath:=false\});\nCalc Mochinvars:=proc(filePATH,p ,polym,m1,m2,a3,n,b,numberofknots,\{cpath: =false\},\{solpath:=false\});" }}{PARA 0 "" 0 "" {TEXT -1 91 "Calc2Moc hinvars:=proc(filePATH,p ,polym,n,b,numberofknots,\{cpath:=false\},\{s olpath:=false\});" }}{PARA 0 "" 0 "" {TEXT -1 268 "Calc3Mochinvars:=pr oc(filePATH,p ,polym,m1,m2,a3,numberofknots,\{cpath:=false\},\{solpath :=false\});\nCalcDihinvars:=proc(filePATH,p,numberofKnots,\{cpath:=fal se\},\{solpath:=false\});\nCalc2twistinvars(filePATH,p,polym,qord,qnum ,numberofknots,\{cpath:=false\},\{solpath:=false\});." }}{PARA 0 "" 0 "" {TEXT -1 920 "These procedures will calculate the values of the inv ariants, generate a file, and write the values to that file. There are options for writing the valid colorings and/ or the solutions of the \+ cocycle to a file also. The first parameter is common to all of these \+ procedures. filePATH is the path to the directory where you wish to sa ve the files for the invariants. This is of type string. numberofknots is also a common parameter. All of these procedures start by calculat ing the value of the trefoil knot. These procedures will then calculat e the values for all knots up to numberofknots. For example, numberofk nots=83 will calculate the values for all knots with 9 and fewer cross ings (248 for knots with 10 a fewer crossings, 800 for 11 and fewer, a nd 2976 for 12 and fewer). Any other place you whish to stop in that r ange can also be chosen. The other common parameter cpath and solpath \+ will be explained below. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 50 " \+ The untwisted case has three procedures " }{TEXT -1 890 "Calc Untwistinvars:=proc(filePATH,p ,qord,qnum,numberofknots,\{cpath:=false \},\{solpath=false\}), Calc2Untwistinvars:=proc(filePATH,p,qord,qnum,n umberofknots,\{cpath=false\},\{solpath:=false\}), and Calc3Untwistinva rs:=proc(filePATH,p,qord,qnum,numberofknots,\{cpath=false\},\{solpath: =false\}). These are for calculating both 2 and 3 cocycle case, 2 cocy cle case only, and 3 cocycle case only, respectively. All of these req uire the same parameters. The prime modulus for the coefficient group \+ is, p. Recall that the quandle in the file \"Quandle\" were stored wit h names Qi_j where i was the size of the quandle and j was the order t hat it appeared in the list. the parameter qord corresponds to i and q num corresponds to j. Note: this restricts these three procedures to o nly work with the quandles Q4_1 to Q6_25 from the file \"Quandle\". By altering these procedure you can obtain other possibilities. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 444 " I wish to organize all of the files containing invariants in the directory /home/csmudde/Inv ariants/ on my machine. I have also created two subdirectories here to store color and solution files (to be described later). I will assign this path to the variable MyDirectory as a string so I can avoid rewr iting it throughout these examples. This directory must exist on your \+ machine. These procedures will NOT create a directory for you." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "#change this to a valid dire ctory on your machine!!!\nMyDirectory:=\"/home/csmudde/Invariants/\"; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,MyDirectoryGQ:/home/csmudde/Inv ariants/6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 504 "For demonstratio n purposes, I will use system calls to show you the contents of the di rectories and files. You may wish to skip these system calls and just \+ use your favorite text editor and file manager to se the results. I w ill comment these out, so if you wish to execute the system calls you \+ will need to alter the paths and uncomment these statements\nls is a L inux command to list files and subdirectories in the specified directo ry. You will see the two subdirectories that I created and no files. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "#uncomment and alter fi lePATH\ntempHolder:=ssystem(\"ls /home/csmudde/Invariants\"):tempHolde r[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q9colorings|+old|+solutions|+ 6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 146 "Now I will calculate the untwisted 2 and 3 cocycle invariants for the quandle Q5_4 with coeffi cient group Z_2 for the first 10 knots in the list. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "CalcUntwistinvars(MyDirectory,2 ,5,4,10,f alse,false);" }}{PARA 6 "" 1 "" {TEXT -1 53 "/home/csmudde/Invariants/ Z_2untwistedQ5_4_2cocinv.inv" }}{PARA 6 "" 1 "" {TEXT -1 53 "/home/csm udde/Invariants/Z_2untwistedQ5_4_3cocinv.inv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/co2solproccallG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q3 finished~all~knots6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 398 "There \+ should now be two files in the directory you specified. They will have the names that were printed above. Everyone of these procedures will \+ print the file name that it creates and writes the data containing the values of the invariants. These files are given a .inv extension. Thi s will make it easier to manage your files. These files are just text \+ files and have a hidden extension as txt. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 95 "#uncomment and alter filePATH\ntempHolder:=ssystem( \"ls /home/csmudde/Invariants\"):tempHolder[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q]pcolorings|+old|+solutions|+Z_2untwistedQ5_4_2cocinv. inv|+Z_2untwistedQ5_4_3cocinv.inv|+6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 194 "The files for the untwisted cases are named with the co efficient group first the word untwisted, the quandle Qqord_qnum, then an underscore with the case 2cocinv or 3cocinv. The two procedures \+ " }{TEXT -1 93 "Calc2Untwistinvars() and Calc3Untwistinvars() are simi lar, but they only calculate one case. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 380 "Next we will use the cat function to print the contents of the file for the 3-cocycle case that was created in the previous e xample. A header is created for any file that these programs create. T his header should contain enough information to allow the user to dete rmine the contents of the file. The knot name , braid word used, and f inally the value of the invariant are listed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "tmpcat:=ssystem(\"cat /home/csmudde/Invariants/Z _2untwistedQ5_4_3cocinv.inv\"):tmpcat[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Qhhn#Untwisted~Quandle~3-cocycle~invariants~for~the~Qua ndle~Q5_4~and~|+#coefficient~group~Z_2|+|+#The~cayley~table~for~the~qu andle~Q5_4|+|+#~~~~~~0~~~~~~0~~~~~~0~~~~~~0~~~~~~0~~|+#~~~~~~1~~~~~~1~ ~~~~~4~~~~~~2~~~~~~3~~|+#~~~~~~2~~~~~~3~~~~~~2~~~~~~4~~~~~~1~~|+#~~~~~ ~3~~~~~~4~~~~~~1~~~~~~3~~~~~~2~~|+#~~~~~~4~~~~~~2~~~~~~3~~~~~~1~~~~~~4 ~~|+#~~|+|+#Generated~Tue~Oct~11~14:01:55~EDT~2005|+|+3_1|*[1,~1,~1]|* ~|+25+12*u^(t[18]+t[24]+t[23])+48*u^(t[123]+t[120]+t[114])|+|+4_1|*[1, ~-2,~1,~-2]|*~|+25+12*u^(t[18]+t[24]+t[23])+48*u^(t[123]+t[115]+t[112] +t[99])|+|+5_1|*[1,~1,~1,~1,~1]|*~|+25|+|+5_2|*[1,~1,~1,~2,~-1,~2]|*~| +25|+|+6_1|*[1,~1,~2,~-1,~-3,~2,~-3]|*~|+25|+|+6_2|*[1,~1,~1,~-2,~1,~- 2]|*~|+25|+|+6_3|*[1,~1,~-2,~1,~-2,~-2]|*~|+25|+|+7_1|*[1,~1,~1,~1,~1, ~1,~1]|*~|+25|+|+7_2|*[1,~1,~1,~2,~-1,~2,~3,~-2,~3]|*~|+25+12*u^(t[18] +t[24]+t[23])+48*u^(t[112]+t[123]+t[99]+t[115])|+|+7_3|*[1,~1,~1,~1,~1 ,~2,~-1,~2]|*~|+25+12*u^(t[18]+t[24]+t[23])+48*u^(t[112]+t[123]+t[99]+ t[115])|+|+7_4|*[1,~1,~2,~-1,~2,~2,~3,~-2,~3]|*~|+25|+|+#Generated~Tue ~Oct~11~14:02:00~EDT~2005|+|+6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 72 " The Mochizuki polynomial cases also contain three proced ures, " }{TEXT -1 367 "CalcMochinvars:=proc(filePATH,p ,polym,m1,m2,a3 ,n,b,numberofknots,\{cpath:=false\},\{solpath:=false\}), Calc2Mochinva rs:=proc(filePATH,p ,polym,n,b,numberofknots,\{cpath:=false\},\{solpat h:=false\}), and Calc3Mochinvars:=proc(filePATH,p ,polym,m1,m2,a3,numb erofknots,\{cpath:=false\},\{solpath:=false\}). The parameters are the same as described in section IV of this worksheet. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 330 " This example will calculate the val ues of the invariants for the first 10 knots where the quandle and coe fficient group are Z_2[t,t^(-1)]/(t^2-t+1) The 2-cocycle formula is f( x,y)=(x-y)^2 * y so we assign the values n1=1, b2=1. The3-cocycle form ula is f(x,y,z)=(x-y)^ * (y-z)^2 so we assign the values m1=0, m2=1, \+ a3=0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "CalcMochinvars(MyD irectory,2,t^2-t+1,0,1,0,1,1,10,false,false);" }}{PARA 6 "" 1 "" {TEXT -1 54 "/home/csmudde/Invariants/Z_2mod(t^2-t+1)2cocinv1_1.inv" } }{PARA 6 "" 1 "" {TEXT -1 56 "/home/csmudde/Invariants/Z_2mod(t^2-t+1) 3cocinv0_1_0.inv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q3finished~all~kno ts6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 510 " These proced ures will calculate the values for the invariant only if the Gcd(g(t), Delta(K)) mod p <> 1 where g(t) is the polynomial used for the quandl e and Delta(K) is the Alexander polynomial for the knot K (stored in K not[i,3]). This is motivated by Inoue's theorem. You should now see th e two above named files in your directory. These files are name by the quandle, then the case 2 cocycle or 3 cocycle by 2coc or 3 coc, and f inally by the exponents that you choose for the Mochizuki formulas. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 271 "A header is created for any i nvariant file that these procedures creates. It should contain all the information to allow you to distinguish the case that was calculated. The knot number, the braid word used, and the value of the invariant \+ appear for each knot calculated." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 111 "The 5 element Dihedral quandle for the first 10 knots with the Mo chizuki cocycle formula for Dihedral quandles." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "CalcDihinvars(MyDirectory,5,10,false,false);" }}{PARA 6 "" 1 "" {TEXT -1 50 "/home/csmudde/Invariants/MochDih_R_53co c_inZ_5.inv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q3finished~all~knots6\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 51 "A system call to see the con tents of our directory." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 " #uncomment and alter filePATH\ntempHolder:=ssystem(\"ls /home/csmudde/ Invariants\"):tempHolder[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Qeucol orings|+MochDih_R_53coc_inZ_5.inv|+old|+solutions|+Z_2mod(t^2-t+1)2coc inv1_1.inv|+Z_2mod(t^2-t+1)3cocinv0_1_0.inv|+Z_2untwistedQ5_4_2cocinv. inv|+Z_2untwistedQ5_4_3cocinv.inv|+6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 195 "The twisted example for the first 10 knots with coeffic ient group Z_2[t,t^(-1)]/(t^3+t+1) with quandle Q4_1. In this example \+ you use qord and qnum to specific the quandle from the file \"Quandle. \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Calc2twistinvars(MyDi rectory,2,t^3+t+1,4,1,10,false,false);" }}{PARA 6 "" 1 "" {TEXT -1 61 "/home/csmudde/Invariants/Z_2modt^3+t+1twistedQ4_1_2cocinv.inv" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q3finished~all~knots6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 90 "Now we should have 5 different files con taining values for invariants for different cases." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "#uncomment and alter filePATH\ntempHolder:= ssystem(\"ls /home/csmudde/Invariants\"):tempHolder[2];" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#Qjwcolorings|+MochDih_R_53coc_inZ_5.inv|+old|+so lutions|+Z_2mod(t^2-t+1)2cocinv1_1.inv|+Z_2mod(t^2-t+1)3cocinv0_1_0.in v|+Z_2modt^3+t+1twistedQ4_1_2cocinv.inv|+Z_2untwistedQ5_4_2cocinv.inv| +Z_2untwistedQ5_4_3cocinv.inv|+6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 1069 " The second to last parameter in these procedures i s for the color files. The last parameter is for the solution files. T hese parameter do NOT have a default value, so if you do not wish to u se these features, you need to pass, false, to the procedure in its po sition in the argument list. These parameters allow the user to specif y a path to a directory where the color data or solution data can be w ritten. If you wish to use these features you can pass the path to th e procedure as follows: \"/home/csmudde/Invariants/colorings/\" for th e path to the color files or \"/home/csmudde/Invariants/solutions/\" f or the path to the solution files. The files containing the valid colo rings is given the extension .clr and the solutions files are given a \+ .sol extension. Both of these are really just text files. The data is \+ written in a way that will allow the user to read the data into a Mapl e worksheet as Maple input. One can then use that data for further cal culations avoiding the overhead of recalculating the data (ie. recalcu lating the valid color vectors). " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 158 "This example will write the solutions to the cocycle conditio n to a file. I will create a variable to store the path where I wish t o write the solution files." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "MySolutions:=\"/home/csmudde/Invariants/solutions/\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,MySolutionsGQD/home/csmudde/Invariants/solut ions/6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 39 "Note this directory \+ is currently empty." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "#un comment and alter filePATH\ntempHolder:=ssystem(\"ls /home/csmudde/Inv ariants/solutions/\"):tempHolder[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#Q!6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 276 "I will now set solpa th=MySolutions. solpath is a optional keyword parameter that was initi alized to false. Note: we do NOT use the assignment operator \":=\". N ow solpath will not evaluate to false and the solutions file will be c reated and written to the directory you specify." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Calc2Mochinvars(MyDirectory,3,t^2+1,1,1,10,fa lse,MySolutions);" }}{PARA 6 "" 1 "" {TEXT -1 65 "/home/csmudde/Invari ants/solutions/coc2SolnsZ_3mod(t^2+1)_1_1.sol" }}{PARA 6 "" 1 "" {TEXT -1 52 "/home/csmudde/Invariants/Z_3mod(t^2+1)2cocinv1_1.inv" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q3finished~all~knots6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 49 "Lets see the contents of the solutions d irectory." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "#uncomment an d alter filePATH\ntempHolder:=ssystem(\"ls /home/csmudde/Invariants/so lutions/\"):tempHolder[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q@coc2So lnsZ_3mod(t^2+1)_1_1.sol|+6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 48 "Now lets see the contents of this solution file." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 137 "#uncomment and alter filePATH\ntempHolder:= ssystem(\"cat /home/csmudde/Invariants/solutions/coc2SolnsZ_3mod(t^2+1 )_1_1.sol\"):tempHolder[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q!6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 440 "Notice how the header file an d time stamps are commented out. This will allow use to read this data into a Maple session. All of these procedures have the ability to wri te the solutions to a file. This option is most useful in the twisted \+ and untwisted cases since the cost associated in solving the cocycle c onditions is high. The Mochizuki polynomial cases can be calculated va ry fast; however, testing the solutions requires some time. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 86 "We now look at an example where w e write the valid colorings of the braids to a file. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "MyColorfiles:=\"/home/csmudde/Invar iants/colorings/\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-MyColorfiles GQD/home/csmudde/Invariants/colorings/6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 34 "This directory is currently empty." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "#uncomment and alter filePATH\ntempHolder:=s system(\"ls /home/csmudde/Invariants/colorings\"):tempHolder[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q!6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 266 "In this example we will use the 3 element Dihedral quan dle with the Mochizuki formula for cocycles for the first ten knots. T his formula is for 3 cocycles, but the colorings for the regions are n ot written to the file. Only the colorings of the braids are written. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CalcDihinvars(MyDirect ory,3,10,MyColorfiles,false);" }}{PARA 6 "" 1 "" {TEXT -1 50 "/home/cs mudde/Invariants/MochDih_R_33coc_inZ_3.inv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q3finished~all~knots6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "#uncomment and alter filePATH\ntempHolder:=ssystem( \"ls /home/csmudde/Invariants/colorings\"):tempHolder[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#QbyColorDih_R_3K3_1.clr|+ColorDih_R_3K4_1.clr|+ ColorDih_R_3K5_1.clr|+ColorDih_R_3K5_2.clr|+ColorDih_R_3K6_1.clr|+Colo rDih_R_3K6_2.clr|+ColorDih_R_3K6_3.clr|+ColorDih_R_3K7_1.clr|+ColorDih _R_3K7_2.clr|+ColorDih_R_3K7_3.clr|+ColorDih_R_3K7_4.clr|+6\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 444 "The colorings for a given braid \+ are written to its own file. Since a small quandle was used, these fil es are small (less than 10 KB). When larger quandles are used and the braid has many valid colorings with that quandle, the size of the col or file can increase (50MB or more). These files are also created so t hat they could be read into a Maple worksheet for use with different c ocycles thus eliminating the cost of finding valid colorings. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "#uncomment and alter filePA TH\ntempHolder:=ssystem(\"cat /home/csmudde/Invariants/colorings/Color Dih_R_3K4_1.clr\"):tempHolder[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#Q afo#The~valid~color~vectors~for~the~braids~colored~with~the~quandle~R_ 3|+#Generated~Tue~Oct~11~14:02:29~EDT~2005|+|+Col4_1R_3[0]:=3;|+Col4_1 R_3[1][1][1]:=0;|+Col4_1R_3[1][1][2]:=0;|+Col4_1R_3[1][1][3]:=0;|+Col4 _1R_3[1][2][1]:=0;|+Col4_1R_3[1][2][2]:=0;|+Col4_1R_3[1][2][3]:=0;|+Co l4_1R_3[1][3][1]:=0;|+Col4_1R_3[1][3][2]:=0;|+Col4_1R_3[1][3][3]:=0;|+ Col4_1R_3[1][4][1]:=0;|+Col4_1R_3[1][4][2]:=0;|+Col4_1R_3[1][4][3]:=0; |+Col4_1R_3[1][5][1]:=0;|+Col4_1R_3[1][5][2]:=0;|+Col4_1R_3[1][5][3]:= 0;|+Col4_1R_3[2][1][1]:=1;|+Col4_1R_3[2][1][2]:=1;|+Col4_1R_3[2][1][3] :=1;|+Col4_1R_3[2][2][1]:=1;|+Col4_1R_3[2][2][2]:=1;|+Col4_1R_3[2][2][ 3]:=1;|+Col4_1R_3[2][3][1]:=1;|+Col4_1R_3[2][3][2]:=1;|+Col4_1R_3[2][3 ][3]:=1;|+Col4_1R_3[2][4][1]:=1;|+Col4_1R_3[2][4][2]:=1;|+Col4_1R_3[2] [4][3]:=1;|+Col4_1R_3[2][5][1]:=1;|+Col4_1R_3[2][5][2]:=1;|+Col4_1R_3[ 2][5][3]:=1;|+Col4_1R_3[3][1][1]:=2;|+Col4_1R_3[3][1][2]:=2;|+Col4_1R_ 3[3][1][3]:=2;|+Col4_1R_3[3][2][1]:=2;|+Col4_1R_3[3][2][2]:=2;|+Col4_1 R_3[3][2][3]:=2;|+Col4_1R_3[3][3][1]:=2;|+Col4_1R_3[3][3][2]:=2;|+Col4 _1R_3[3][3][3]:=2;|+Col4_1R_3[3][4][1]:=2;|+Col4_1R_3[3][4][2]:=2;|+Co l4_1R_3[3][4][3]:=2;|+Col4_1R_3[3][5][1]:=2;|+Col4_1R_3[3][5][2]:=2;|+ Col4_1R_3[3][5][3]:=2;|+#Generated~Tue~Oct~11~14:02:29~EDT~2005|+|+|+| +|+6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 258 "This example only had the trivial colorings. Files containing nontriv ial colorings can become very long. Once again, using this cat functio n is for demonstration proposes only. Use you text editor to view the \+ contents of these files. It will save you time!! " }}}{EXCHG {PARA 0 " " 0 "" {TEXT 200 951 "Final Notes. These programs do not allow one to \+ pass the valid coloring to them. This will be a feature in a future up date. This will be a nice feature because it will allow use to examine how different cocycles effect the value of the invariant for a given \+ braid and quandle. We will not have the exponential cost of evaluating the colorings since you will pass them to the procedure. We will also have to write the C program to calculate the colorings of the regions . The C program for calculating the valid colorings of the braids allo wed use to calculate with larger quandles than expected. So the assump tion that Maple could color the regions fast was false. This will also be a great update! Programs for calculating other Alexander quandles \+ may also be included. This package is currently limited to work only w ith primes p. The updates made here allow use to calculate many, many more values so this is a good stopping point for this update. " }}}} {MARK "195" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 137522212 137901640 134779940 1095924708 } {RTABLE M7R0 I6RTABLE_SAVE/137522212X,%)anythingG6"6"[gl$!%"!!#*!#!#!#"#"!"!#F& } {RTABLE M7R0 I6RTABLE_SAVE/137901640X*%)anythingG6"6"[gl$!%"!!"$!#"""F& } {RTABLE M7R0 I6RTABLE_SAVE/134779940X*%)anythingG6"6"[gl$!%"!!"\cadct![cadct!!!!!!!!!"!!##"" !#!!""##!!""##!!"""""""""#""!!##"!##""!!#"##!!""#########!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 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