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1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle12" -1 226 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 234 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 225 "" 0 "" {TEXT 218 61 "Worksheet III for quand le 2-cocycle and 3-cocycle invariants " }}{PARA 225 "" 0 "" {TEXT 218 26 "Mochizuki Polynomial Cases" }}{PARA 212 "" 0 "" {TEXT 213 32 "by M asahico Saito & Chad Smudde " }}{PARA 228 "" 0 "" {TEXT 227 0 "" }}} {EXCHG {PARA 215 "" 0 "" {TEXT 204 166 " These programs compute \+ quandle cocycle invariants of classical knots where the two or three c ocycles are Mochizuki polynomial cocycles for Alexander quandles. " }} }{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 8 "restart;" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 18 "ALEXANDER QUANDLES" }}{PARA 206 "" 0 " " {TEXT 208 633 "First, we will look at Alexander quandles. In the pre vious worksheets, the quandles that we used were taken from the file \+ \"Quandle.\" The table of quandles that were used for this file came f rom the book, ``Surfaces in 4-space,'' by Carter, Kamada, and Saito, S pringer-Verlag, 2004. We now are going to look at a special type of qu andle called Alexander quandles. Procedures to generate the Cayley ta bles for these quandles will be explained in the first part of this wo rksheet, and procedures that calculate the 2 and 3-cocycle invariants \+ where the cocycles are Mochizuki polynomial cocycles will be explained in the second part. " }}}{EXCHG {PARA 228 "" 0 "" {TEXT 205 60 "Step \+ 1: Read the file \"alexQuanpkg.m\" into the worksheet. " }}{PARA 228 " " 0 "" {TEXT 205 92 " Determine where the file \"alexQuanpkg .m\" is saved and make a note of the path. " }}{PARA 228 "" 0 "" {TEXT 205 143 " Next use the read command followed by double quot es and the path where the file was saved. e.g. [> read \"path to alexQ uanpkg.m\"; ." }{TEXT 205 2 "\n" }{TEXT 205 61 " Here are so me examples that are platform specific." }{TEXT 205 2 "\n" }{TEXT 205 85 " Windows: [>. read \"C:\\\\maple.local\\\\Pro ject\\\\Top.txt\". " }{TEXT 205 2 "\n" }{TEXT 205 84 " UNIX: [> read \"/usr/local/maple.local/Project/Top.txt\"." }{TEXT 205 2 "\n" }{TEXT 205 87 " Macintosh: [> re ad \"Main HD:Local Maple Files:Project:Top.txt\" " }{TEXT 205 2 "\n" }{TEXT 205 165 " If the files are saved in the same director y (folder) and that directory is currently the active directory, then \+ you may only need to type the file name, " }{TEXT 205 2 "\n" }{TEXT 205 42 " such as [> read ``filename'' . " }{TEXT 205 2 "\n" }{TEXT 205 133 " For more information on reading the file in to the worksheet type ?read or ?file at the prompt to see Maples onli ne help. " }{TEXT 205 2 "\n" }{TEXT 205 136 " If all else fa ils download the file \"AlexQuandlepkgprocs.mws\" and use this as a te mplate for these and future calculations." }{TEXT 205 2 "\n" }{TEXT 205 69 " The procedures included in the \"alexQuanpkg.m\" fi le are:" }{TEXT 205 2 "\n" }{TEXT 205 44 " quan dlesize(Quande)," }{TEXT 205 2 "\n" }{TEXT 205 46 " \+ Alexmult(a,b,polym,p)," }{TEXT 205 2 "\n" }{TEXT 205 38 " \+ rres(polym,p)," }{TEXT 205 2 "\n" }{TEXT 205 39 " \+ findelt(L,pol)," }{TEXT 205 2 "\n" }{TEXT 205 45 " \+ AlexQuandle(polym,p)." }{TEXT 205 2 "\n" }{TEXT 205 142 " The procedure AlexQuandle() is the main procedu re that will be used. The other local procedures will be discussed in \+ less detail." }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 1 "r" } {MPLTEXT 1 216 22 "ead \"alexQuanpkg.m\";" }}{PARA 234 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 334 "Recall th at 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 operation a*b=ta+(1-t)b is a finite quandle. The pr ocedure AlexQuandle() will generate the Cayley table for this quandle \+ and return an isomorphic quandle whose Cayley table has elements in \{ 0..n-1\} where n is the order of the quandle." }{TEXT 208 2 "\n" } {TEXT 208 153 "This will allow us to reuse old programs that are set t o only accept quandles that are zero indexed two dimensional arrays wi th elements in \{0..n-1\}. " }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 29 "Aquandle:=AlexQuandle(t+1,3):" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 32 "print(convert(Aquandle,matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'matrixG6$%*protectedGI(_syslibG6\"6#7%7%\"\"!\"\"# \"\"\"7%F,F-F+7%F-F+F," }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 85 "Sinc e p is prime and h(t)=a_0+a_1*t+...+a_n*t^n is a polynomial (in t), e lements of " }{TEXT 208 39 " Z_p[t,t^-1]/(h(t)) are represented by " } {TEXT 208 58 "the reduced residue system, i.e., polynomials of the for m " }{TEXT 208 49 "b_0+b_1*t+...+b_(n-1)*t^(n-1) where b_i is in Z_p" }{TEXT 208 722 ". 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 expr essed by a polynomial as above, and every element is represented by a \+ polynomial of a positive degree. Then by taking the remainder when div ided by h(t), every element is represented by a polynomial as above. T he purpose of the local procedure rres() is to list all such polynomia ls. Then the procedure Alexmult() will multiply all possibilities a*b \+ where a, b are in the reduced residue system. The resulting polynomial will then be reduced back into the reduced residue system. Then the p rocedure findelt() will find that polynomial in the list of reduced re sidues and return that integer value. " }}}{EXCHG {PARA 228 "" 0 "" {TEXT 205 48 "Step 2: Read the braid words into the worksheet." }{TEXT 205 2 "\n" }{TEXT 205 458 " The file \"knotsLivingston.txt \" contains the knot name, braid words, and Alexander polynomials for the prime knots in the table up to and including 12 crossings. The kn ots are organized as follows. The array name is \"Knot.\" There are 29 77 such prime knots according to Charles Livingston's knot database. T he array \"Knot\" is then first numbered from 0 to 2966. The second in dex 1,2, or 3 then represents the knot name , braid word, and the " } {TEXT 205 36 "Alexander polynomial, respectively. " }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 29 "read \"knotsLivingston.txt\":" }}} {EXCHG {PARA 206 "" 0 "" {TEXT 208 64 "The following example returns t he knot name of the trefoil knot." }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 10 "Knot[0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q$3_16 \"" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 43 "Next we get the braid wo rd for the trefoil." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 10 " Knot[0,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"F#F#" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 75 "Finally, we get the Alexander polynomi al from the third element or the row." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 10 "Knot[0,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I \"tG6\"\"\"#\"\"\"F(F%!\"\"F(F(" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 39 "MOCHIZUKI POLYNOMIAL COCYCLE INVARIANTS" }}{PARA 206 "" 0 "" {TEXT 208 57 "First we need to read the procedures into the worksheet. " }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 25 "read \"PolycocInvp kg.m\";" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 56 "The following proce dures are included into the package: " }}{PARA 206 "" 0 "" {TEXT 208 70 " Invariants2_3Mochizuki(Xpolym,Apolym,p,m1,m2,a3,n1,b2,Kn ot)," }{TEXT 208 2 "\n" }{TEXT 208 33 " findgood3cocexp(g,p,n )," }{TEXT 208 2 "\n" }{TEXT 208 54 " Invar2Mochizuki(Xpolym, Apolym,p,m1,a2,Knot)," }{TEXT 208 2 "\n" }{TEXT 208 46 " cocy cle2check(cocy,Xpolym,Apolym,p)," }{TEXT 208 2 "\n" }{TEXT 208 47 " \+ Mochizuki2coc(Xpolym,Apolym,p,m1,a2)," }{TEXT 208 2 "\n" }{TEXT 208 16 " Xi(n)," }{TEXT 208 2 "\n" }{TEXT 208 38 " \+ TableInvars(n,p,g,m1,m2,a3)," }{TEXT 208 2 "\n" }{TEXT 208 57 " \+ Invar3Mochizuki(Xpolym,Apolym,p,m1,m2,a3,Knot)," }{TEXT 208 2 "\n" }{TEXT 208 27 " makeinv(Quandle)," }{TEXT 208 2 "\n" }{TEXT 208 46 " cocycle3check(cocy,Xpolym,Apolym,p)," }{TEXT 208 2 " \n" }{TEXT 208 50 " Mochizuki3coc(Xpolym,Apolym,p,m1,m2,a3)," }{TEXT 208 2 "\n" }{TEXT 208 35 " MochizukiDihedral3coc(p)," }{TEXT 208 2 "\n" }{TEXT 208 50 " CalcDihinvars(filePATH,p,n umberofKnots)," }{TEXT 208 2 "\n" }{TEXT 208 67 " Calcinvars( filePATH,p ,polym,n,b,m1,m2,a3,numberofKnots)," }{TEXT 208 2 "\n" } {TEXT 208 27 " ismult(charstar)," }{TEXT 208 2 "\n" }{TEXT 208 74 " calc3cocInvar(Quandle,Knot,m::posint,(optional) cocy cle values)," }{TEXT 208 2 "\n" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 205 994 "Let X=Z_p[t,t^-1]/((h(t)) be an Alexander quandle and A=Z_p[t,t^- 1]/(g(t)) be an Alexander quandle used for the coefficient group. Here g(t) must divide h(t) mod p. Following Mochizuki, we compute that fo r x_i in X, f(x_1,x_2,...,x_n)=(x_1-x_2)^(p^m1)*(x_2-x_3)^(p^m2)*...*( x_(n-1)-x_n)^(p^m_(n-1))*x_n^(a_n) is a cocycle where p is prime and e ither a_n is 0 or a power of p such that g(t) divides 1-t^((p^m1)+(p^m 2)+...+a_n) mod p. Mochizuki2coc:=proc(Xpolym,Apolym,p,m1,a2) and Moch izuki3coc:=proc(Xpolym,Apolym,p,m1,m2,a3) are procedures that will gen erate the values for the 2 and 3-cocycle cases respectively. All possi ble elements are taken from the quandle X and the value for f(x,y) (or f(x,y,z)) are calculated and reduced mod p and mod g(t). This happens only if either a_n is zero or if a_n<>0 and g(t) divides 1-t((p^m1)+( p^m2)+...+a_n) mod p. These procedures will either return a table of v alues for the cocycle or will print Mochizuki conditions not satisfied and return false. " }}{PARA 206 "" 0 "" {TEXT 208 40 "First an exampl e in the 2-cocycle case. " }{TEXT 208 718 "Mochizuki2coc:=proc(Xpolym, Apolym,p,m1,a2) takes as its first and second arguments the polynomial s for the quandles X and A (Xpolym=h(t) and Apolym=g(t)). The third ar gument is a prime p used for both quandles and the cocycle formula. Th e argument m1 is the exponent of p for the first term of the product( \+ (x_1-x_2)^(p^m1) ) and a2 is the exponent for the last term, x_2. The \+ argument a_2 can be zero or a power of p. In the case that a power of p is desired, the complete term must be passed not just the power of \+ p (eg. a_2=p^3). In the case that the final term is a prime power, the procedure does not test that it is indeed a prime power, but it only \+ checks that g(t) divides 1- t^((p^m1)+(p^m2)+...+a_n) mod p." }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 47 "ex1_2coc:=Mochizuki2coc( t^2-t+1,t^2-t+1,3,2,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)ex1_2cocG6 \"I\"fGF$" }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 16 "print(ex1_ 2coc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&ARRAYG%*protectedG6$7$;\"\" !\"\")F'7]p/6$F(F(F(/6$F(\"\"\"\"\"#/6$F(F0F//6$F(\"\"$F//6$F(\"\"%F(/ 6$F(\"\"&F0/6$F(\"\"'F0/6$F(\"\"(F//6$F(F)F(/6$F/F(F//6$F/F/F(/6$F/F0F 0/6$F/F5F0/6$F/F8F//6$F/F;F(/6$F/F>F(/6$F/FAF0/6$F/F)F//6$F0F(F0/6$F0F /F//6$F0F0F(/6$F0F5F(/6$F0F8F0/6$F0F;F//6$F0F>F//6$F0FAF(/6$F0F)F0/6$F 5F(F0/6$F5F/F//6$F5F0F(/6$F5F5F(/6$F5F8F0/6$F5F;F//6$F5F>F//6$F5FAF(/6 $F5F)F0/6$F8F(F(/6$F8F/F0/6$F8F0F//6$F8F5F//6$F8F8F(/6$F8F;F0/6$F8F>F0 /6$F8FAF//6$F8F)F(/6$F;F(F//6$F;F/F(/6$F;F0F0/6$F;F5F0/6$F;F8F//6$F;F; F(/6$F;F>F(/6$F;FAF0/6$F;F)F//6$F>F(F//6$F>F/F(/6$F>F0F0/6$F>F5F0/6$F> F8F//6$F>F;F(/6$F>F>F(/6$F>FAF0/6$F>F)F//6$FAF(F0/6$FAF/F//6$FAF0F(/6$ FAF5F(/6$FAF8F0/6$FAF;F//6$FAF>F//6$FAFAF(/6$FAF)F0/6$F)F(F(/6$F)F/F0/ 6$F)F0F//6$F)F5F//6$F)F8F(/6$F)F;F0/6$F)F>F0/6$F)FAF//6$F)F)F(" }}} {EXCHG {PARA 228 "" 0 "" {TEXT 205 137 "In the above example, the term a_2 was zero so the quandle A was not restricted at all. The followin g example has term a_2 a prime power." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 49 "ex2_2coc:=Mochizuki2coc(t^2-t+1,t^2-t+1,3,2,3^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)ex2_2cocG6\"I\"fGF$" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 16 "print(ex2_2coc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&ARRAYG%*protectedG6$7$;\"\"!\"\")F'7]p/6$F(F(F(/ 6$F(\"\"\"\"\"#/6$F(F0F0/6$F(\"\"$F0/6$F(\"\"%F(/6$F(\"\"&F0/6$F(\"\"' F0/6$F(\"\"(F0/6$F(F)F(/6$F/F(F(/6$F/F/F(/6$F/F0F//6$F/F5F//6$F/F8F(/6 $F/F;F(/6$F/F>F(/6$F/FAF//6$F/F)F(/6$F0F(F(/6$F0F/F//6$F0F0F(/6$F0F5F( /6$F0F8F(/6$F0F;F//6$F0F>F//6$F0FAF(/6$F0F)F(/6$F5F(F(/6$F5F/F//6$F5F0 F(/6$F5F5F(/6$F5F8F(/6$F5F;F//6$F5F>F//6$F5FAF(/6$F5F)F(/6$F8F(F(/6$F8 F/F0/6$F8F0F0/6$F8F5F0/6$F8F8F(/6$F8F;F0/6$F8F>F0/6$F8FAF0/6$F8F)F(/6$ F;F(F(/6$F;F/F(/6$F;F0F//6$F;F5F//6$F;F8F(/6$F;F;F(/6$F;F>F(/6$F;FAF// 6$F;F)F(/6$F>F(F(/6$F>F/F(/6$F>F0F//6$F>F5F//6$F>F8F(/6$F>F;F(/6$F>F>F (/6$F>FAF//6$F>F)F(/6$FAF(F(/6$FAF/F//6$FAF0F(/6$FAF5F(/6$FAF8F(/6$FAF ;F//6$FAF>F//6$FAFAF(/6$FAF)F(/6$F)F(F(/6$F)F/F0/6$F)F0F0/6$F)F5F0/6$F )F8F(/6$F)F;F0/6$F)F>F0/6$F)FAF0/6$F)F)F(" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 109 "In this case the procedure performs a divides test to \+ make sure the choices satisfy the Mochizuki conditions." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 36 "Divide(1-t^(3^2+3^3),t^2-t+1) m od 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 187 "The following is an example where the final term is nonzero and the Mochizuki conditions are not satisfied. Here the polynomial for the Alexander quandle A=Z_p[t,t^-1]/(g(t)) wa s changed." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 49 "ex3_2coc: =Mochizuki2coc(t^2-t+2,t^2-t+2,3,2,3^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "QCMochizuki~conditions~not~satisfied6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)ex3_2cocG6\"I&falseG%*protectedG" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 58 "The next statement shows where the previous exam ple fails." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 36 "Divide(1- t^(3^2+3^3),t^2-t+2) mod 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG% *protectedG" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 205 78 "The 3-cocycle c ase is similar except that there are more terms in the product." }}} {EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 49 "ex4_3coc:=Mochizuki3coc( t^2-t+1,t^2-t+1,2,2,1,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)ex4_3coc G6\"I\"fGF$" }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 16 "print(ex 4_3coc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&ARRAYG%*protectedG6$7%;\" \"!\"\"$F'F'7\\o/6%F(F(F(F(/6%F(F(\"\"\"F(/6%F(F(\"\"#F(/6%F(F(F)F(/6% F(F/F(F//6%F(F/F/F(/6%F(F/F2I\"tG6\"/6%F(F/F),&F;F/F/F//6%F(F2F(F//6%F (F2F/F?/6%F(F2F2F(/6%F(F2F)F;/6%F(F)F(F//6%F(F)F/F;/6%F(F)F2F?/6%F(F)F )F(/6%F/F(F(F(/6%F/F(F/F//6%F/F(F2F?/6%F/F(F)F;/6%F/F/F(F(/6%F/F/F/F(/ 6%F/F/F2F(/6%F/F/F)F(/6%F/F2F(F;/6%F/F2F/F//6%F/F2F2F(/6%F/F2F)F?/6%F/ F)F(F?/6%F/F)F/F//6%F/F)F2F;/6%F/F)F)F(/6%F2F(F(F(/6%F2F(F/F;/6%F2F(F2 F//6%F2F(F)F?/6%F2F/F(F?/6%F2F/F/F(/6%F2F/F2F//6%F2F/F)F;/6%F2F2F(F(/6 %F2F2F/F(/6%F2F2F2F(/6%F2F2F)F(/6%F2F)F(F;/6%F2F)F/F?/6%F2F)F2F//6%F2F )F)F(/6%F)F(F(F(/6%F)F(F/F?/6%F)F(F2F;/6%F)F(F)F//6%F)F/F(F;/6%F)F/F/F (/6%F)F/F2F?/6%F)F/F)F//6%F)F2F(F?/6%F)F2F/F;/6%F)F2F2F(/6%F)F2F)F//6% F)F)F(F(/6%F)F)F/F(/6%F)F)F2F(/6%F)F)F)F(" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 205 752 "One should observe the values that the 3-cocycle examp le above has. In the beginning of this worksheet we generated the Alex ander quandle and created a isomorphic quandle whose elements were in \+ \{0.. n-1\}. These are the values that are used for coloring the braid s. This same isomorphism is used in generating the cocycle values. The arguments to the Mochizuki cocycle formula are in the set \{0..n-1\} \+ and then calculations of the Mochizuki formula use the polynomial that represents the integer. The formula is then reduced in the Alexander \+ quandle A=Z_p[t,t^-1]/(g(t)) and stays in polynomial form. These value s in A will carry over to the state sum and be seen in the value of th e invariant. More about this later when that procedure is explained." }}}{EXCHG {PARA 228 "" 0 "" {TEXT 205 386 "The above two procedures do not check that the values calculated actually satisfy the 2 or 3 cocy cle conditions. There are special local functions for this. They are c ocycle3check:=proc(cocy,Xpolym,Apolym,p) and cocycle2check:=proc(cocy, Xpolym,Apolym,p). The last three parameters are the same as described \+ above. The first parameter is the table of values returned from the pr ocedures " }{TEXT 205 232 "Mochizuki2coc:=proc(Xpolym,Apolym,p,m1,a2) \+ or Mochizuki3coc:=proc(Xpolym,Apolym,p,m1,m2,a3) . This procedure subs titutes these values into the cocycle conditions and checks that they \+ vanish. The return value is either true or false." }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 42 "cocycle2check(ex2_2coc,t^2-t+1,t^2-t+1,3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 42 "cocycle3check(ex4_3coc,t^2-t+1, t^2-t+1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {EXCHG {PARA 228 "" 0 "" {TEXT 205 65 "I will now change a value of th e cocycles to show a false output." }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 56 "The value of f (0,0,0) in example 4 before manipulation." }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 16 "ex4_3coc[0,0,0];" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 69 "Now we change the value of one of the terms to create a false output." }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 216 21 "ex4_3coc[0, 0,0]:=t+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I)ex4_3cocG6\"6%\"\"!F'F ',&I\"tGF%\"\"\"F*F*" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 88 "Now we check if the altered table satisfies the cocycle conditions (which it shouldn't)." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 42 "cocycle 3check(ex4_3coc,t^2-t+1,t^2-t+1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 (Q#f[6\"\"\"!F%F%Q#]=F$,&I\"tGF$\"\"\"F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 61 "ex4_3coc[0,0,0]:=0;cocycle3check(ex4_3coc,t^2-t+1,t ^2-t+1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I)ex4_3cocG6\"6%\"\"!F'F 'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 395 "Now we look at calculating the value \+ of the Mochizuki 2 and 3-cocycle invariants with the procedures Invar2 Mochizuki:=proc(Xpolym,Apolym,p,m1,a2,Knot) and Invar3Mochizuki:=proc( Xpolym,Apolym,p,m1,m2,a3,Knot). The only parameter that is different i n these two procedures when compared to the procedures, Mochizuki2coc: =proc(Xpolym,Apolym,p,m1,a2) and Mochizuki3coc:=proc(Xpolym,Apolym,p,m 1,m2,a3)" }{TEXT 208 299 ", calculating the cocycle values is the last parameter which is the braid word. These procedures will calculate th e cocycle values and test if they satisfy the cocycle condition by cal ling the above described procedures. It will then proceed to color the braid and return the value of the invariant." }}}{EXCHG {PARA 228 "> \+ " 0 "" {MPLTEXT 1 216 61 "Knot[0,1];Invar3Mochizuki(t^2+t+1,t^2+t+1,2, 0,1,0,Knot[0,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q$3_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"#;\"\"\"*&\"#[F$)I\"uG6\"I\"tGF)F$F$" }}} {EXCHG {PARA 228 "" 0 "" {TEXT 205 619 "One can see that the coefficie nt group is Z_2[t,t^-1]/(t^2+t+1) which has the reduced residue system \{0,1,t,1+t\}. Recall that the procedure calculating the values for t he cocycles left the value in polynomial form. The polynomial form is \+ carried through to the output of the invariant. Therefore, the \"t\" i n the above example represents an element of \{0,1,t,1+t\}. The value \+ of the invariant shows that there were 48 colorings that took the valu e of t in \{0,1,t,1+t\}. One must be careful here since in the first w orksheet t (really t was an array) represented a free variable. The 2- cocycle invariant is similar." }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 476 "Given the complexity of coloring the braid with the quandle, we m ay wish to get the most out of our calculation. Since the shadow color ing case uses only those colorings of the braid that are \"valid\" we \+ could calculate a 3-cocyle invariant at the same time as the two cocyc le invariant with the small added cost of calculating the colorings of the regions of only the \"valid\" braid colorings. The procedure Inva riants2_3Mochizuki:=proc(Xpolym,Apolym,p,m1,m2,a3,n1,b2,Knot)" }{TEXT 208 2 "\n" }{TEXT 208 233 "will do this. The first six parameters are \+ the same as the 3-cocycle invariant procedure. Now the variables n1 an d b1 represent the exponents for the Mochizuki 2-cocycle formula. Fin ally, a braid word is given as the last parameter." }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 171 "A special procedure Xi(n) is given to produ ce the alternating polynomial in t Sum((-t)^i,i = 0 .. n-1). This is u seful for creating polynomials for the Alexander quandles." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 10 "T3:=Xi(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#T3G6\",(*$)I\"tGF$\"\"#\"\"\"F*F(!\"\"F*F*" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 62 "Knot[0,1];Invariants2_3Mochizuk i(T3,T3,5,0,1,0,2,0,Knot[0,2]);" }{MPLTEXT 1 201 2 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q$3_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$D',,F #\"\"\"*&\"%]PF%)I\"uG6\",&I\"tGF*F%\"\"$F%F%F%*&F'F%)F),&*&\"\"%F%F,F %F%\"\"#F%F%F%*&F'F%)F),&*&F-F%F,F%F%F2F%F%F%*&F'F%)F),&*&F3F%F,F%F%F% F%F%F%" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 128 "The first value ret urned, 625 in this example, represents the value of the 2-cocycle inva riant, and the second element returned," }{XPPEDIT 18 0 "Typesetting:- mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"625\") , Typesetting:-mo(\"+\", form = \"infix\", fence = \"false\", separato r = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspac e\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mrow(Typesetting:-mn(\"3750\"), Typesetting:-mo(\"⁢\", \+ form = \"infix\", fence = \"false\", separator = \"false\", lspace = \+ \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", font_style_name = \"2D Commen t\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255, 255]\"), Typesetting:-msup(Typesetting:-mi(\"u\"), Typesetting:-mrow(T ypesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", st retchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", mins ize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typ esetting:-mi(\"t\"), Typesetting:-mo(\"+\", form = \"infix\", fence = \+ \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", m axsize = \"infinity\", minsize = \"1\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", font_style_name = \"2D Comment\" , size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255 ]\"), Typesetting:-mn(\"3\")), Typesetting:-mo(\")\", form = \"postfix \", fence = \"true\", separator = \"false\", lspace = \"thinmathspace \", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\")), superscriptshift = \"0\"), Typesetting:-mi(\"\")) , Typesetting:-mo(\"+\", form = \"infix\", fence = \"false\", separato r = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspac e\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mrow(Typesetting:-mn(\"3750\"), Typesetting:-mo(\"⁢\", \+ form = \"infix\", fence = \"false\", separator = \"false\", lspace = \+ \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", font_style_name = \"2D Commen t\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255, 255]\"), Typesetting:-msup(Typesetting:-mi(\"u\"), Typesetting:-mrow(T ypesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", st retchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", mins ize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typ esetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"4\"), Typesett ing:-mo(\"⁢\", form = \"infix\", fence = \"false\", sep arator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"f alse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0 ]\", background = \"[255,255,255]\"), Typesetting:-mi(\"t\")), Typeset ting:-mo(\"+\", form = \"infix\", fence = \"false\", separator = \"fal se\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stre tchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsi ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"2\" )), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separ ator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmaths pace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\")), superscrip tshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"+\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"mediu mmathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symm etric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgro und = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mn(\"3750\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"fal se\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stret chy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-msup(Types etting:-mi(\"u\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"p refix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetti ng:-mrow(Typesetting:-mn(\"3\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \+ \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", font_style_name = \"2D Commen t\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255, 255]\"), Typesetting:-mi(\"t\")), Typesetting:-mo(\"+\", form = \"infi x\", fence = \"false\", separator = \"false\", lspace = \"mediummathsp ace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mn(\"4\")), Typesetting:-mo(\")\", fo rm = \"postfix\", fence = \"true\", separator = \"false\", lspace = \" thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", \+ symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", font_sty le_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", bac kground = \"[255,255,255]\")), superscriptshift = \"0\"), Typesetting: -mi(\"\")), Typesetting:-mo(\"+\", form = \"infix\", fence = \"false\" , separator = \"false\", lspace = \"mediummathspace\", rspace = \"medi ummathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Comment\", size = \+ \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mrow(Typesetting:-mn(\"3750\"), Typesetting:-mo(\"&Invisible Times;\", form = \"infix\", fence = \"false\", separator = \"false\", \+ lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"u\"), Typesett ing:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", \+ separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmath space\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infin ity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"2\" ), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"f alse\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", str etchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mins ize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"t\" )), Typesetting:-mo(\"+\", form = \"infix\", fence = \"false\", separa tor = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathsp ace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting :-mn(\"1\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"tru e\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"ver ythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize \+ = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Comment\", size \+ = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), \+ superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mi(\" \")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6,-I#mnGF$6#Q$625F'-I#moGF$63Q\"+F'/ %%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF>/%'lspaceGQ0mediummat hspaceF'/%'rspaceGFC/%)stretchyGF>/%*symmetricGF>/%(maxsizeGQ)infinity F'/%(minsizeGQ\"1F'/%(largeopGF>/%.movablelimitsGF>/%'accentGF>/%0font _style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+ba ckgroundGQ.[255,255,255]F'-F#6&-F26#Q%3750F'-F663Q1⁢F'F 9F " 0 "" {MPLTEXT 1 201 23 "R3:=Al exQuandle(t+1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#R3G6\"I(QuandleG F$" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 26 "print(convert(R3, matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'matrixG6$%*protectedGI(_ syslibG6\"6#7%7%\"\"!\"\"#\"\"\"7%F,F-F+7%F-F+F," }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 343 "The Mochizuki 3-cocycle formula is given by f[i ,j,k]:=((i-j)*((2*k^p-j^p)-(2*k-j)^p)/p) mod p. The procedure Mochizuk iDihedral3coc:=proc(p) will generate the values for the 3-cocycle. In \+ this case the prime for the Mochizuki 3-cocycle formula and quandle R3 are the same. Therefore, only the prime number needs to be passed to \+ the procedure. " }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 33 "M3D coc:=MochizukiDihedral3coc(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'M3D cocG6\"I\"fGF$" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 14 "print (M3Dcoc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&ARRAYG%*protectedG6$7%; \"\"!\"\"#F'F'7=/6%F(F(F(F(/6%F(F(\"\"\"F(/6%F(F(F)F(/6%F(F/F(F(/6%F(F /F/F(/6%F(F/F)F//6%F(F)F(F(/6%F(F)F/F//6%F(F)F)F(/6%F/F(F(F(/6%F/F(F/F //6%F/F(F)F)/6%F/F/F(F(/6%F/F/F/F(/6%F/F/F)F(/6%F/F)F(F(/6%F/F)F/F)/6% F/F)F)F(/6%F)F(F(F(/6%F)F(F/F)/6%F)F(F)F//6%F)F/F(F(/6%F)F/F/F(/6%F)F/ F)F)/6%F)F)F(F(/6%F)F)F/F(/6%F)F)F)F(" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 318 "The procedure, calc3cocInvar:=proc(Quandle,Knot,m::posi nt), from worksheet #1 (the untwisted worksheet) can be used to calcul ate the invariant. Recall that this procedure had an optional 4th argu ment that would take the table of cocycle values. So in this case we j ust use the above procedures to get the desired case." }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 201 47 "Knot[0,1];calc3cocInvar(R3,Knot [0,2],3,M3Dcoc);" }{MPLTEXT 1 201 2 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q$3_16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"*\"\"\"*&\"#=F$I\" uG6\"F$F$" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 208 128 "The next workshe et will look at procedures that help in doing mass calculations in a e ffective and effortless (for the user) way" }}}{PARA 224 "" 0 "" {TEXT 228 0 "" }}{PARA 202 "" 0 "" {TEXT 229 0 "" }}{PARA 226 "" 0 "" {TEXT 230 0 "" }}{PARA 223 "" 0 "" {TEXT 231 0 "" }}{PARA 210 "" 0 "" {TEXT 232 0 "" }}{PARA 217 "" 0 "" {TEXT 233 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }