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{SECT 0 {EXCHG {PARA 209 "" 0 "" {TEXT 210 46 "\nWorksheet II for quan
dle 2-cocycle invariants" }}{PARA 210 "" 0 "" {TEXT 211 32 "by Masahic
o Saito & Chad Smudde " }}{PARA 211 "" 0 "" }}{EXCHG {PARA 212 "" 0 ""
 {TEXT 212 162 "        These programs compute quandle cocycle invaria
nts of classical knots for the ``twisted case\" using 2-cocycles,  see
 the reference below for more details. " }}{PARA 212 "" 0 "" {TEXT 
212 270 "This worksheet is a continuation to the worksheet \"cocysampl
e.mws\" which deals with the \"untwisted case\". See the worksheet \"c
ocysample.mws\" for more information about how to read in files and fo
r explanations of the contents of the files \"knot_table\" and \"Quand
le\".    " }}{PARA 212 "" 0 "" {TEXT 212 104 "        The general theo
ry for which this worksheet and package are based is presented in, for
 example, " }}{PARA 212 "" 0 "" {TEXT 212 90 "the paper,  ``Quandle Ho
mology Theory and Cocycle Knot Invariants,'' by Carter and Saito, " }}
{PARA 212 "" 0 "" {TEXT 212 214 "Proc. in Symposia in Pure Math., vol.
 71, Topology and Geometry of Manifolds, eds. Matic and McCrory, AMS (
2003), 249-268, and the book ``Surfaces in 4-space,'' by Carter, Kamad
a, and Saito, Springer-Verlag, 2004. " }}{PARA 212 "" 0 "" {TEXT 212 
468 "        Acknowledgement: Some part of the programs written by Dan
 Jelsovski earlier was used. Prof. Edwin Clark helped us with solving \+
equations over finite cyclic abelian groups. Conversations with collab
orators in this project on quandle cocycle invariants were helpful. Th
e collaborators include J.S. Carter, M. Elhamdadi, M. Grana, A. Harris
, D. Jelsovski, S. Kamada, L. Langford, M.A. Nikiforou, S. Satoh. This
 project is partially supported by NSF DMS-0301089.  " }}}{EXCHG 
{PARA 211 "> " 0 "" {MPLTEXT 1 213 8 "restart;" }}}{EXCHG {PARA 211 ""
 0 "" {TEXT 214 55 "Step 1: Read the file \"KnotpkgT.m\" into the work
sheet. " }{TEXT 214 65 "\n           The procedures included in the \"
KnotpkgT.m\" file are:" }{TEXT 214 81 "\n                        co2Tw
istedSol:=proc(Quandle,polynomial (in t),m::posint)" }{TEXT 214 133 "
\n                        co2TwistedInvar:=proc(Quandle,Knot,polynomia
l (in t), m::posint, (optional)solutions)                       " }
{TEXT 214 49 "\n                        CollectTerms:=proc(List)" }
{TEXT 214 67 "\n                        tinverse=proc(polynomial (in t
),m::posint)" }{TEXT 214 51 "\n                        quandlesize:=pr
oc(Quandle)" }{TEXT 214 48 "\n                        makeinv:=proc(Qu
andle)." }{TEXT 214 153 "\n             The procedures makinv(), quand
lesize(), and tinverse() are procedures that are used in the others an
d are of little importance to the user." }{TEXT 214 60 "\n            \+
 The other procedures will be described below. " }}}{EXCHG {PARA 211 "
> " 0 "" {MPLTEXT 1 213 18 "read \"KnotpkgT.m\":" }}}{EXCHG {PARA 211 
"> " 0 "" {MPLTEXT 1 213 18 "read \"knot_table\":" }}}{EXCHG {PARA 
211 "> " 0 "" {MPLTEXT 1 213 15 "read \"Quandle\":" }}}{EXCHG {PARA 
211 "" 0 "" {TEXT 214 142 "The following examples show how braid words
, quandles, and lists can be used to compute quandle 2-cocycle invaria
nts for the \"twisted case.\"  " }}}{EXCHG {PARA 213 "" 0 "" {TEXT 
215 181 "The first step is to find solutions to the \"twisted\" 2-cocy
cle condition. The procedure \"co2TwistedSol(Quandle,polynomial (in t)
, modulus)\" takes as input the following three items:" }{TEXT 215 
104 "\n         1) Quandle is a two dimensional zero indexed array tha
t will later be used to color the braid." }{TEXT 215 127 "\n         2
) Polynomial (in t) This is the polynomial h(t) from the Alexander qua
ndle, Z_m[t,t^(-1)]/(h(t)), that will be used." }}{PARA 213 "" 0 "" 
{TEXT 215 78 "         3) The modulus, m, from the Alexander quandle Z
_m[t, t^(-1)]/(h(t)). " }}{PARA 213 "" 0 "" {TEXT 215 159 "           \+
     Note: It would be best to use only a prime modulus for the time b
eing. Some of the built in Maple procedures that are used in \"co2Twis
tedInvar\"" }{TEXT 215 86 "\n                         require a prime \+
modulus even though no testing is performed." }}{PARA 213 "" 0 "" 
{TEXT 215 528 "The output of this procedure is a two dimensional array
 representing the solutions to the \"twisted\" 2-cocycle conditions. T
his is in fact a general solution. Thus, there are most likely free va
riables in the solution. They are represented by x[i] and take values \+
from Z_m. Upon assigning values to these free variables one can obtain
 a specific \"twisted\" 2-cocycle. Otherwise, if desired the solution \+
with these free variables can be sent to the procedure \"co2TwistedInv
ar\" and will appear in the output (to be described later)." }}}
{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 216 30 "f:=co2TwistedSol(Q4_3,t+
1,2); " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"F$" }{TEXT 217 1 "
 " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 216 9 "print(f);" }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#-I&ARRAYGI*protectedGF%6$7%;\"\"!\"\"$F(;F)
F)72/6%F)F)F)F)/6%F)\"\"\"F),&&I\"xG6\"6#F1F1&F46#F*F1/6%F)\"\"#F),&F3
F1&F46#F;F1/6%F)F*F),&F7F1&F46#\"\"%F1/6%F1F)F),*F3F1F=F1F7F1FBF1/6%F1
F1F)F)/6%F1F;F)F3/6%F1F*F)F7/6%F;F)F),&F3F1FBF1/6%F;F1F),(F3F1F=F1FBF1
/6%F;F;F)F)/6%F;F*F)FB/6%F*F)F),&F7F1F=F1/6%F*F1F),(F=F1F7F1FBF1/6%F*F
;F)F=/6%F*F*F)F)" }{TEXT 217 1 " " }}}{EXCHG {PARA 213 "> " 0 "" 
{MPLTEXT 1 216 33 "g:=co2TwistedSol(Q4_3,t^2+t+1,2);" }}{PARA 214 "" 
1 "" {XPPMATH 20 "6#>I\"gG6\"I\"fGF%" }{TEXT 217 1 " " }}}{EXCHG 
{PARA 213 "> " 0 "" {MPLTEXT 1 216 9 "print(g);" }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#-I&ARRAYGI*protectedGF%6$7%;\"\"!\"\"$F(;F)\"\"\"7B/6%F
)F)F)F)/6%F)F)F,F)/6%F)F,F),&*&,(&I\"xG6\"6#\"\"(F,&F86#\"\")F,&F86#\"
\"%F,F,I\"tGF9F,F,&F86#\"\"#F,/6%F)F,F,*&,*&F86#F*F,F?F,&F86#\"\"'F,F7
F,F,FBF,/6%F)FEF),&*&,&FLF,F?F,F,FBF,F,&F86#F,F,/6%F)FEF,*&,,F?F,&F86#
\"\"&F,FLF,F7F,F<F,F,FBF,/6%F)F*F),(*&,*FLF,FZF,FJF,F<F,F,FBF,F,FTF,FC
F,/6%F)F*F,*&FLF,FBF,/6%F,F)F),&*&,(F7F,FLF,FZF,F,FBF,F,FCF,/6%F,F)F,*
&,.FJF,F?F,FZF,FLF,F7F,F<F,F,FBF,/6%F,F,F)F)/6%F,F,F,F)/6%F,FEF),(*&,&
FJF,FLF,F,FBF,F,FTF,FCF,/6%F,FEF,*&F?F,FBF,/6%F,F*F)FT/6%F,F*F,*&F7F,F
BF,/6%FEF)F),&*&,&FJF,F?F,F,FBF,F,FTF,/6%FEF)F,*&,&FLF,FZF,F,FBF,/6%FE
F,F),(*&,(F?F,FZF,F7F,F,FBF,F,FTF,FCF,/6%FEF,F,*&FJF,FBF,/6%FEFEF)F)/6
%FEFEF,F)/6%FEF*F)FC/6%FEF*F,*&F<F,FBF,/6%F*F)F),(F5F,FTF,FCF,/6%F*F)F
,*&,(FJF,F?F,FLF,F,FBF,/6%F*F,F),&F_pF,FTF,/6%F*F,F,*&,&F?F,F<F,F,FBF,
/6%F*FEF),&*&,*FLF,FZF,F<F,F?F,F,FBF,F,FCF,/6%F*FEF,*&FZF,FBF,/6%F*F*F
)F)/6%F*F*F,F)" }{TEXT 217 1 " " }}}{EXCHG {PARA 213 "" 0 "" {TEXT 
215 545 "The procedure \"co2TwistedInvar\" will call the procedure \"c
o2TwistedSol.\" Hence a call to \"co2TwistedSol\" executed by the user
 might be an uncommon occurrence. It could be desirable in a couple of
 circumstances. First if one wished to study the different cocycles th
at can be obtained from a given quandle and Alexander quandle, or seco
ndly, if one wishes to save the solutions for a given quandle and Alex
ander quandle so that the solutions could be sent as an optional input
 to the procedure \"co2TwistedInvar\" in order to speed up calculation
s." }}}{EXCHG {PARA 213 "" 0 "" {TEXT 215 55 "Now the procedure \"co2T
wistedInvar\" will be explained. " }{TEXT 215 35 "\nThis procedure wil
l take as input:" }}{PARA 213 "" 0 "" {TEXT 215 384 "           1) A q
uandle. This is the quandle that will be used to color the braid. Any \+
quandle from the file \"Quandle\" can be used, or any two dimensional \+
zero indexed                                                          \+
                                       array representing the Cayley t
able for the quandle operation that satisfies the three quandle proper
ties could be used." }{TEXT 215 114 "\n           2) A knot. A minimum
 braid from any of the files, \"knot_table\", \"extknot_table\", or ov
erextKnotTable\"." }{TEXT 215 84 "\n           3) The polynomial h(t) \+
from the Alexander quandle Z_m[t, t^(-1)]/(h(t))." }{TEXT 215 113 "\n \+
          4) The modulus m from the Alexander quandle Z_m[t, t^(-1)]/(
h(t)). Let m be prime for the time being." }}{PARA 213 "" 0 "" {TEXT 
215 568 "This procedure will check all possible colorings and when a v
alid coloring is found the Boltzman weight of the coloring is found an
d added to the list. Therefore, the output from the \"co2TwistedInvar
\" is a list [B(C)] for C in the valid colorings of the braid by the g
iven quandle where B stands for Boltzman weight. Note: B(C), the Boltz
man weight for a particular coloring, is in the Alexander quandle Z_m[
t, t^(-1)]/(h(t)), so if no values were assigned to the free variables
 B(C) may be a family of elements from Z_m[t, t^(-1)]/(h(t)) for a par
ticular coloring C. " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 213 34 
"co2TwistedInvar(Q4_3,bw[1],t+1,2);" }}{PARA 214 "" 1 "" {XPPMATH 20 "
6#72\"\"!,(&I\"xG6\"6#\"\"$\"\"\"&F'6#F+F+&F'6#\"\"#F+F%F%F%F$F%F%F%F%
F$F%F%F%F%F$" }{TEXT 217 1 " " }}}{EXCHG {PARA 215 "" 0 "" {TEXT 218 
277 "The procedure \"CollectTerms\" can be applied to the output of \"
co2TwistedInvar\" to obtain an easily recognizable list of Boltzman we
ights. The elements of this list are two element lists. The first elem
ent is the number of occurrences of the second element, the Boltzman w
eight." }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 213 48 "CollectTerms(
co2TwistedInvar(Q4_3,bw[1],t+1,2));" }}{PARA 214 "" 1 "" {XPPMATH 20 "
6#7$7$\"\"%\"\"!7$\"#7,(&I\"xG6\"6#\"\"$\"\"\"&F+6#F/F/&F+6#\"\"#F/" }
{TEXT 217 1 " " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 214 114 "By creating
 lists of quandles and Alexander quandles many different combinations \+
of calculations can be performed." }}{PARA 213 "" 0 "" {TEXT 215 359 "
This first example will calculate the list of Boltzman weights for the
 Alexander quandle Z_2[t,t^(-1)]/(t+1) and for the three and four elem
ent quandles listed in the file \"Quandle.\" These quandles will be pu
t into a list and the nested loops will allow us to perform all possib
le calculations. A complete list of the quandles stored in the file \"
Quandle\" is " }{TEXT 215 194 "L:=[Q3,Q4_1,Q4_2,Q4_3,Q5_1,Q5_5,Q5_2,Q5
_3,Q5_4,Q5_6,Q6_1,Q6_2,Q6_3,Q6_4,Q6_5,Q6_6,Q6_7,Q6_8,Q6_9,Q6_10,Q6_11,
Q6_12,Q6_13,Q6_14,Q6_15,Q6_16,Q6_17,Q6_18,Q6_19,Q6_20,Q6_21,Q6_22,Q6_2
3,Q6_24,Q6_25]." }{TEXT 215 1 "\n" }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 213 23 "L:=[Q3,Q4_1,Q4_2,Q4_3]:" }{MPLTEXT 1 213 14 "\nfor \+
i in L do" }{MPLTEXT 1 213 30 "\n  print(convert(i,matrix));  " }
{MPLTEXT 1 213 24 "\n  for j from 1 to 35 do" }{MPLTEXT 1 213 27 "\n  \+
  print(Knot, j, bw[j]);" }{MPLTEXT 1 213 57 "\n    print(CollectTerms
(co2TwistedInvar(i,bw[j],t+1,2)));" }{MPLTEXT 1 213 6 "\n  od;" }
{MPLTEXT 1 213 4 "\nod;" }{MPLTEXT 1 213 4 "\n   " }}{PARA 214 "" 1 ""
 {XPPMATH 20 "6#-I'matrixG6$I*protectedGF&I(_syslibG6\"6#7%7%\"\"!\"\"
#\"\"\"7%F-F.F,7%F.F,F-" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"\"\"7%F%F%F%" }{TEXT 217 1 " " }}{PARA 214 
"" 1 "" {XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}{PARA 214 ""
 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"#7&\"\"\"!\"#F'F(" }{TEXT 217 1 " "
 }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }
}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"$7'\"\"\"F'F'F'F'" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"%7(\"
\"\"F'\"\"#F(!\"\"F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 
20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 
20 "6%I%KnotG6\"\"\"&7)!\"\"\"\"#F'\"\"$!\"#F)F(" }{TEXT 217 1 " " }}
{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}
{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"'7(!\"\"\"\"#F'F(F(F(" 
}{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"(7(!
\"\"\"\"#F(F'F'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6
#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I
%KnotG6\"\"\")7)\"\"\"F'F'F'F'F'F'" }{TEXT 217 1 " " }}{PARA 214 "" 1 
"" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 ""
 {XPPMATH 20 "6%I%KnotG6\"\"\"*7+!\"\"\"\"$F(F(\"\"#\"\"\"F*!\"$F)" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#57*\"
\"\"F'\"\"#!\"\"F(F(F(F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#67+\"\"\"F'\"\"#\"\"$F)!\"\"F(!\"$F(" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"*\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#77*\"
\"\"F'F'F'\"\"#!\"\"F(F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#87+\"\"\"!\"#!\"\"F)\"\"$\"\"#F+F+F*" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#97+\"
\"\"!\"$\"\"#F(F)!\"\"F)F(F)" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#:7,!\"\"\"\"#\"\"$!\"#F'\"\"%F+F)F(!\"%" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#;7*!\"
\"\"\"#F(F(F(F(F'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 
"6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6
%I%KnotG6\"\"#<7,!\"\"F'!\"#\"\"\"\"\"%F*\"\"$!\"%F(F+" }{TEXT 217 1 "
 " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " "
 }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#=7+\"\"\"F'F'\"\"$!\"
#!\"$F*F'F)" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$
\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%Knot
G6\"\"#>7*\"\"\"F'F'!\"#F'F'F'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 ""
 {XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#?7+!\"\"\"\"#F'!\"$F(F(F(\"\"$F*" }{TEXT 
217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 
217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#@7*\"\"\"F'F
'F'!\"#F(F'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7
$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%Kno
tG6\"\"#A7+!\"\"\"\"#\"\"\"F)!\"$F(F(F*F*" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#B7*!\"\"\"\"#F'F'F'F(F(F(" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#C7*!\"
\"\"\"#F(F'F'F(F(F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 
"6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6
%I%KnotG6\"\"#D7+!\"\"\"\"#F(!\"$F(\"\"$F*F'F(" }{TEXT 217 1 " " }}
{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}
{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#E7,\"\"\"!\"#\"\"$!\"%F)
F*\"\"#F'!\"$F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7
#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%K
notG6\"\"#F7+\"\"\"F'\"\"#!\"$F(!\"\"F)F)F(" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#G7+\"\"\"F'\"\"#F(!\"\"!\"$F(F
*F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"
!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#H7
+\"\"\"F'!\"#F'\"\"$F)\"\"#F*F)" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#I7*\"\"\"F'!\"#F'F'F(F'F(" }{TEXT 217 1 " \+
" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"!" }{TEXT 217 1 " " 
}}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#J7*!\"\"\"\"#F'F(F(F'F
'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"
!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#K7
*\"\"\"!\"#F'F(F'F(F'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"#F\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#L7*\"\"\"\"\"#F'F(F'F(F(F'" }{TEXT 217 1 "
 " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " "
 }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#M7*\"\"\"F'F'\"\"#!\"
\"F)F)F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"
*\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"
\"#N7*\"\"\"!\"#F(F'F'\"\"#F)F)" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"\"*\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#-I'matrixG6$I*protectedGF&I(_syslibG6\"6#7&7&\"\"!\"\"#
\"\"\"F,7&F-F.F,F.7&F.F,F-F-7&\"\"$F2F2F2" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"\"7%F%F%F%" }{TEXT 217 1 " " 
}}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"#5\"\"!" }{TEXT 217 1 " " }}
{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"#7&\"\"\"!\"#F'F(" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"%\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"$7'\"
\"\"F'F'F'F'" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$
\"\"%\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%Knot
G6\"\"\"%7(\"\"\"F'\"\"#F(!\"\"F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "
" {XPPMATH 20 "6#7#7$\"\"%\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"\"&7)!\"\"\"\"#F'\"\"$!\"#F)F(" }{TEXT 217 
1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"#5\"\"!" }{TEXT 217 1 "
 " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"'7(!\"\"\"\"#F'F(F
(F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"%\"\"
!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"(
7(!\"\"\"\"#F(F'F'F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 
20 "6#7#7$\"\"%\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 
20 "6%I%KnotG6\"\"\")7)\"\"\"F'F'F'F'F'F'" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"%\"\"!" }{TEXT 217 1 " " }}{PARA 
214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"\"*7+!\"\"\"\"$F(F(\"\"#\"\"\"F
*!\"$F)" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"%
\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"
\"#57*\"\"\"F'\"\"#!\"\"F(F(F(F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 ""
 {XPPMATH 20 "6#7#7$\"\"%\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#67+\"\"\"F'\"\"#\"\"$F)!\"\"F(!\"$F(" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"#5\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#77*\"
\"\"F'F'F'\"\"#!\"\"F(F(" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"\"%\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#87+\"\"\"!\"#!\"\"F)\"\"$\"\"#F+F+F*" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7#7$\"\"%\"\"!" }
{TEXT 217 1 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6%I%KnotG6\"\"#97+\"
\"\"!\"$\"\"#F(F)!\"\"F)F(F)" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#7#7$\"#5\"\"!" }{TEXT 217 1 " " }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6%I%KnotG6\"\"#:7,!\"\"\"\"#\"\"$!\"#F'\"\"%F+F)F(!\"%" }
{TEXT 217 1 " " }}{PARA 216 "" 1 "" {TEXT 219 33 "Warning,  computatio
n interrupted" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 215 149 "The next exa
mple shows how loops and lists could be used to color with a specific \+
quandle and have different Alexander quandles for the coefficients." }
}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 216 17 "p:=[t+1,t^2+t+1]:" }
{MPLTEXT 1 216 15 "\nfor i  in p do" }{MPLTEXT 1 216 20 "\n  for j in \+
[2,3] do" }{MPLTEXT 1 216 32 "\n  print(Z_,j,[t,t^(-1)],\"/\",i);" }
{MPLTEXT 1 216 24 "\n  for k from 1 to 35 do" }{MPLTEXT 1 216 25 "\n  \+
  print(Knot,k,bw[k]);" }{MPLTEXT 1 216 56 "\n    print(CollectTerms(c
o2TwistedInvar(Q3,bw[k],i,j)));" }{MPLTEXT 1 216 5 "\n    " }{MPLTEXT 
1 216 10 "\nod;od;od;" }}{PARA 213 "> " 0 "" }}{EXCHG {PARA 213 "> " 0
 "" }}{PARA 217 "" 0 "" }{PARA 218 "" 0 "" }{PARA 219 "" 0 "" }{PARA 
220 "" 0 "" }{PARA 221 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }