M7R0
I+Calcinvarsf*6+%)filePATHG%"pG%&polymG%"nG%"bG%#m1G%#m2G%#a3G%.numberofKnotsG6
+%"iG%#HHG%-tmpfilename1G%-tmpfilename2G%-tmpfilename3G%&tmpf1G%&tmpf2G%&tmpf3G
%'INVARSG6"F8C4>8&-%$catG629$%#Z_G9%%"mG%"oG%"dGQ"(F8-%(convertG6$-%%evalG6#9&%
'stringGQ")F8""#%'cocinvG9'%"_G9(Q".F8%$txtG>8'-F=64F?F@FAFBFCFDFEFFFN""$FP9)FR
9*FR9+FTFU>F;-&%,StringToolsG6#%'RemoveG6$%'ismultGF;>FW-Fjn6$F_oFW-%&printG6#F
;-Fdo6#FW>8)-%&fopenG6$F;%&WRITEG>8*-F[p6$FWF]p-%(fprintfG6$FioQWMochizuki~2-co
cycle~invariants~for~Alexander~Quandles|+F8-Fcp6$F_pQWMochizuki~3-cocycle~invar
iants~for~Alexander~Quandles|+F8-Fcp6'FioQM2-cocycle~formula~f(x,y)=(x-y)^%d^%d
~*y^%d~|+F8FAFQFS-Fcp6)F_pQen3-cocycle~formula~f(x,y,z)=(x-y)^%d^%d~*(y-z)^%d^%
d~*z^%d|+F8FAFenFAFfnFgn-Fcp6%FioQ6Alexander~Quandle~%s|+F8-F=6)F@FAQ+[t^1,t^-1
]F8Q"/F8FEFFFN-Fcp6%F_pFaqFbq?(8$""!"""9,%%trueG@$0-%$modG6$-%$GcdG6$FL&%%KnotG
6$FiqFZFAF[rC%>8,-%7Invariants2_3MochizukiG6+FLFLFAFenFfnFgnFQFS&Fgr6$FiqFO-Fcp
6+FioQB%s|*%s|*~Gcd(%s,%s)~mod~%d~=%s|+%s|+|+F8-FG6$&Fgr6$FiqF[rFM-FG6$F_sFM-FG
6$FLFM-FG6$FfrFMFA-FG6$F`rFM-FG6$&F[s6#F[rFM-Fcp6+F_pQC%s|*%s|*~Gcd(%s,%s)~mod~
%d~=%s~|+%s|+|+F8FdsFhsFjsF\tFAF^t-FG6$&F[s6#FOFM-%'fcloseG6#Fio-F\u6#F_p-F\u6#
8+F8F8F8F8,
I'ismultf*6#%)charstarGF8F8F8@%/F?Q"*F8OF]rO%&falseGF8F8F8F8,
I7Invariants2_3Mochizukif*6+%'XpolymG%'ApolymGF&F*F+F,%#n1G%#b2GFgr6C%(SST2cocG
%(SST3cocG%+SST2contriG%+SST3contriG%$jj3G%$jj5G%$jj6G%$jj8G%"sG%$numG%&ColorG%
&brindG%%indxG%.ColDiffMatch0G%+ColorDiff0G%%GinvG%-quandleorderG%-co2Solutions
G%-co3SolutionsG%+FacecolorsG%+facecolorsG%-precolorfaceG%)nowcolorG%,colcolori
ngG%(QuandleG%,VALIDCOLORSG%)QuandleCG%&KnotCG%"xG%"yGF/%*valid2cocG%*valid3coc
GF8F8C->FiqFjq>8%Fjq>8CF\v>8DF\v>8<-%,AlexQuandleG6$F?FL>84-%,quandlesizeG6#F^y
>83-%(makeinvGFfy>86-%.Mochizuki3cocG6(F?FAFLFQFSFen>85-%.Mochizuki2cocG6'F?FAF
LFfnFgn@$3-%%typeG6$Faz%&tableG-Fhz6$F\zFjzC$>Fjx-%.cocycle2checkG6&FazF?FAFL>F
\y-%.cocycle3checkG6&F\zF?FAFL@%3FjxF\yC'>8/,&-%$maxG6#-%#opG6#-%$mapG6$f*6#Fax
F86$%)operatorG%&arrowGF8-%$absG6#F?F8F8F8F\rF[rF[rF[r?(8AF[rF[r,&-%%nopsG6#F\r
F[rF[rF[rF]rC$>&8.6#F^]l-%&arrayG6$;F[rFj[l7">&87Fg]l-Fi]l6#;Fjq,&Fj[lF[r!""F[r
>8-)FcyFj[l?(80FjqF[r,&Ff^lF[rFd^lF[rF]rC'?(FioF[rF[rFj[lF]r>&&Ff]lFctF]u-Far6$
-%%iquoG6$Fi^l)Fcy,&FioF[rFd^lF[rFcy?(F_pF[rF[rF`]lF]r@%2Fjq&F\rF_u?(FbuF[rF[rF
j[lF]rC&@$/Fbu-F[]l6#Fj_l>&&Ff]l6#,&F_pF[rF[rF[rFau&&Ff]lF_u6#,&FbuF[rF[rF[r@$/
Fbu,&F_`lF[rF[rF[r>Fb`l&F^y6$&Fg`l6#,&FbuF[rFd^lF[r&Fg`lFau@$2FbuF_`l>Fb`lFcal@
$2F\alFbu>Fb`lFcal?(FbuF[rF[rFj[lF]rC&@$F^`l>Fb`l&Fhy6$FcalFf`l@$F[al>Fb`lF`al@
$Feal>Fb`lFcal@$Fhal>Fb`lFcal>82-%&evalmG6#,&F__lF[r&Ff]l6#F_]lFd^l>81-%$sumG6$
-F[]l6#&Fgbl6#%#jjG/FgclF[^l@$/F_clFjqC%?(88FjqF[r,&FcyF[rFd^lF[rF]rC(>F;Fjq>FW
Fjq?(89F[rF[rF_]lF]r>&&F_^l6#Fcdl6#FjqF]dl?(8:F[rF[rF`]lF]r?(8;FjqF[r,&-F[]l6#&
F\r6#FjdlF[rFd^lF[rF]r>&&F_^lFael6#,&F\elF[rF[rF[r&F^y6$&Fdel6#F\el&&Ff]lFaelFe
el?(F[sF[rF[rF`]lF]r@%2Fjq&F\r6#F[s>FW,&FWF[r&F\z6%&&F_^lFafl6#,&-F[]l6#F`flF[r
Fd^lF[r&&Ff]lFafl6#Fjfl&F]gl6#,&FjflF[rF[rF[rF[r>FW,&FWF[r&F\z6%Fffl&&Ff]l6#,&F
[sF[rF[rF[rF^gl&FgglF`glFd^l>Fhx,&FhxF[r)%"uG-Far6$-%$RemG6%FWFA%"tGFLF[r?(F[sF
[rF[rF`]lF]r@%F_fl>F;,&F;F[r&Faz6$F\glF_glF[r>F;,&F;F[r&Faz6$FfglFjglFd^l>Fiq,&
FiqF[r)F^hl-Far6$-Fbhl6%F;FAFdhlFLF[rO6$FiqFhxC$-Fdo6#Q1Invalid~cocyclesF8OF8F8
F8F8F8,
I0findgood3cocexpf*6%%"gGF&F(6&F*F+%#m3G%"LGF8F8C%>FWF\^l?(FiqFjqF[rFLF]r?(FhxF
jqF[rFLF]r?(F;FjqF[rFLF]r@$-Far6$-%'DivideG6$,&F[rF[r)Fdhl,()FAFiqF[r)FAFhxF[r)
FAF;F[rFd^lF?FA>FW7$-F`\lFgo7'F?FAFiqFhxF;OFWF8F8F8F8,
I0Invar2Mochizukif*6(F`vFavF&F*%#a2GFgr64%$SSTG%*SSTcontriGFivFjvF[wF\wF]wF^wF_
wF`wFawFbwFcwFdwFew%,coSolutionsG%)validcocGF]xF8F8C*>FiqFjq>FcyF\v>FazF_y>Fgbl
-Fey6#Faz>F_cl-FjyFi\m>Fhy-Fcz6'F?FAFLFQFS@$-Fhz6$FhyFjz>Fcy-F`[l6&FhyF?FAFL@%F
cyC'>Ff^l,&-F]\l6#-F`\l6#-Fc\l6$f*Ff\lF8Fg\lF8Fj\lF8F8F8FenF[rF[rF[r?(F;F[rF[r,
&-Fa]l6#FenF[rF[rF[rF]r>&F[sFeo-Fi]l6#;F[rFf^l>Fbu)FgblFf^l?(Ff]lFjqF[rFbalF]rC
(?(FWF[rF[rFf^lF]r>&FbtFgo-Far6$-Fc_l6$Ff]l)Fgbl,&FWF[rFd^lF[rFgbl?(8(F[rF[rFb^
mF]r@%2Fjq&Fen6#Fg_m@%2Fj_m"$+"?(FioF[rF[rFf^lF]rC&@$/Fio-F[]l6#Fj_m>&&F[s6#,&F
g_mF[rF[rF[rF]u&&F[sF[`m6#,&FioF[rF[rF[r@$/Fio,&Fc`mF[rF[rF[r>Ff`m&Faz6$&F[am6#
Ff_l&F[amF]u@$2FioFc`m>Ff`mFfam@$2F`amFio>Ff`mFfam?(FioF[rF[r&Ff^l6#%#KTGF]rC&@
$/Fio,&-F[]l6#&&%#bwGF_bmF[`mF[r!$+"F[r>Ff`mFj`m@$/Fio,&FebmF[r!#**F[r>Ff`mFdam
@$2FioFdbm>Ff`mFfam@$2F^cmFio>Ff`mFfam?(FioF[rF[rFf^lF]rC&@$Fb`m>Ff`m&F_cl6$Ffa
mFj`m@$F_am>Ff`mFdam@$Fham>Ff`mFfam@$F[bm>Ff`mFfam>FhxFjq>Fi^l-Fibl6#,&FbtF[r&F
[s6#Fa^mFd^l>Fj[l-Facl6$-F[]l6#&Fi^lFfcl/FgclFh^m@$/Fj[lFjqC$?(F_pF[rF[rFb^mF]r
@%2Fjq&FenF_u@$2FgemF^`m>Fhx,&FhxF[r&Fhy6$&&F[sF_u6#-F[]l6#Fgem&F_fm6#,&FafmF[r
F[rF[rF[r>Fhx,&FhxF[r&Fhy6$&&F[sFd`lF`fm&F[gmFdfmFd^l>Fiq,&FiqF[r)F^hl-Far6$-Fb
hl6%FhxFAFdhlFLF[rOFiqC$FiilOF8F8F8F8F8,
I.cocycle2checkf*6&%%cocyGF`vFavF&6-F/%"jG%"kGFaxFbx%"zG%"wG%%tempG%(rrespolG%'
quanodGF]xF8F8C(>F[s-%%rresG6$FAFQ>Ff]l-F`yFghm>Ff^l-Fa]lFafl?(FiqFjqF[rFj^lF]r
@$0&F?6$FiqFiqFjqC$-Fdo6'Q#f[F8FiqFiqQ#]=F8F_imOF\v?(FWFjqF[rFj^lF]r?(Fg_mFjqF[
rFj^lF]r?(FioFjqF[rFj^lF]rC$>Fbu,*&F?6$FWFg_mF[r&F?6$&Ff]lF^jmFioF[r&F?6$FWFioF
d^l&F?6$&Ff]lFcjm&Ff]l6$Fg_mFioFd^l@$0-Far6$-Fbhl6%FbuFLFdhlFQFjqC$-Fdo6(Q6temp
~from~ReidIII~is~F8FbuQ'xyz~isF8FWFg_mFioOF\vOF]rF8F8F8F8,
I.Mochizuki2cocf*6'F`vFavF&F*F]\m6*F/F\hm%"fG%&qsizeGF]x%*XelementsGF`hm%*tempc
heckG6#%)rememberGF8C*>FioF\^l>Fio-FfhmFay>FW-Fa]lF]u>Fbu-Far6$-F][m6$,&F[rF[r)
Fdhl,&)FLFQF[rFSF[rFd^lF?FL@%5/FSFjqFbu%)continueGC$-Fdo6#QCMochizuki~condition
s~not~satisfiedF8OF\v>F;-Fi]l6$;FjqFe_mF[^n?(FiqF[rF[rFWF]r?(FhxF[rF[rFWF]rC$>F
_p*&),&&Fio6#FiqF[r&Fio6#FhxFd^lF^]nF[r)Fe^nFSF[r>&F;6$,&FiqF[rFd^lF[r,&FhxF[rF
d^lF[r-Far6$-Fbhl6%F_pFAFdhlFLOF;F8F8F8F8,
I#Xif*6#F(F8F8F8@%/F?FjqOFjqO-Facl6$),$FdhlFd^lF//F/;Fjq,&F?F[rFd^lF[rF8F8F8F8,
I,TableInvarsf*6(F(F&F`jlF*F+F,6$F/%(tempgcdGF8F8?(FiqFjqF[rF?F]rC$>FhxF`r@$0Fh
xF[r-Fdo6&FfsF_sFhx-%0Invar3MochizukiG6)FLFLFAFQFSFenF_sF8F8F8F8,
I0Invar3Mochizukif*6)F`vFavF&F*F+F,Fgr69F_\mF`\mFivFjvF[wF\wF]wF^wF_wF`wFawFbwF
cwFdwFewFa\mFhwFiwFjwF[xF\xFb\mF]xF8F8C*>FiqFjq>FcdlF\v>FjdlF_y>Fgbl-FeyFael>F_
cl-FjyFael>FhyF]z@$F`]m>Fcdl-Fd[lFd]m@%FcdlC'>Ff^l,&-F]\l6#-F`\l6#-Fc\l6$f*Ff\l
F8Fg\lF8Fj\lF8F8F8FfnF[rF[rF[r?(F;F[rF[r,&-Fa]l6#FfnF[rF[rF[rF]rC$>Fe^mFf^m>&Fc
yFeo-Fi]l6#;FjqFj^l>FbuFj^m?(Ff]lFjqF[rFbalF]rC'?(FWF[rF[rFf^lF]r>F__mF`_m?(Fg_
mF[rF[rF\cnF]r@%2Fjq&FfnF[`m?(FioF[rF[rFf^lF]rC&@$/Fio-F[]l6#F]dn>Ff`mFj`m@$/Fi
o,&FbdnF[rF[rF[r>Ff`m&FjdlFcam@$2FioFbdn>Ff`mFfam@$2FgdnFio>Ff`mFfam?(FioF[rF[r
Ff^lF]rC&@$Fadn>Ff`mF[dm@$Ffdn>Ff`mFdam@$F[en>Ff`mFfam@$F^en>Ff`mFfam>Fi^l-Fibl
6#,&FbtF[r&F[s6#F[cnFd^l>Fj[lF[em@$Fbem?(FazFjqF[r,&FgblF[rFd^lF[rF]rC'>FhxFjq?
(F\zF[rF[rF[cnF]r>&&Fcy6#F\zFhdlFaz?(F_^lF[rF[rF\cnF]r?(F]dlFjqF[r,&-F[]l6#&Ffn
6#F_^lF[rFd^lF[rF]r>&&FcyFagn6#,&F]dlF[rF[rF[r&Fjdl6$&Fdgn6#F]dl&&F[sFagnFegn?(
F_pF[rF[rF\cnF]r@%2Fjq&FfnF_u>Fhx,&FhxF[r&Fhy6%&&FcyF_u6#,&-F[]l6#F`hnF[rFd^lF[
r&F_fm6#Fihn&F_fm6#,&FihnF[rF[rF[rF[r>Fhx,&FhxF[r&Fhy6%Fehn&F[gmF\in&F[gmF^inFd
^l>FiqF^gmOFiqC$FiilOF8F8F8F8F8,
I(makeinvf*6#F]x6'F/F\hmF`hmFcjlFewF^\nF8C&>Fg_m-FeyF\]l>FW-Fi]l6%;Fjq,&Fg_mF[r
Fd^lF[rFdjnF\^l?(FiqFjqF[rFejnF]r?(FhxFjqF[rFejnF]rC$>F;&F?Fgil>&FW6$FhxF;FiqOF
WF8F8F8F8,
I.cocycle3checkf*FigmF[hmF8F8C(>F[sFehm>Ff]lFihm>Ff^lF[im?(FiqFjqF[rFj^lF]r?(Fh
xFjqF[rFj^lF]r@$0&F?6%FiqFhxFhxFjqC$-Fdo6(FdimFiqFhxFhxFeimFi[oOF\v?(FWFjqF[rFj
^lF]r?(Fg_mFjqF[rFj^lF]r?(FioFjqF[rFj^lF]r?(F_pFjqF[rFj^lF]rC$>Fbu,.&F?6%FWFioF
_pF[r&F?6%FWFg_mF_pFd^l&F?6%FWFg_mFioF[r&F?6%FajmFioF_pFd^l&F?6%FfjmFgjmF_pF[r&
F?6%&Ff]l6$FWF_p&Ff]l6$Fg_mF_p&Ff]l6$FioF_pFd^l@$FjjmC$-Fdo6)Fb[nFbuQ(xyzw~isF8
FWFg_mFioF_pOF\vOF]rF8F8F8F8,
I.Mochizuki3cocf*6(F`vFavF&F*F+F,6+F/F\hmF]hmFj[nF[\nF]xF\\nF`hmF]\nF^\nF8C*>F_
pF\^l>F_pFc\n>Fg_m-Fa]lF_u>F[s-Far6$-F][m6$,&F[rF[r)Fdhl,(F^]nF[r)FLFSF[rFenF[r
Fd^lFAFL@%5/FenFjqF[sFb]nC$Fd]nOF\v>FW-Fi]l6%FdjnFdjnFdjn?(FiqF[rF[rFg_mF]r?(Fh
xF[rF[rFg_mF]r?(F;F[rF[rFg_mF]rC$>Fbu*(),&&F_pFd^nF[r&F_pFf^nFd^lF^]nF[r),&Fb`o
F[r&F_pFeoFd^lF`_oF[r)Fe`oFenF[r>&FW6%F[_nF\_n,&F;F[rFd^lF[r-Far6$-Fbhl6%FbuFAF
dhlFLOFWF8F8F8F8,
I6MochizukiDihedral3cocf*6#F&Fb^oF^\nF8C%>FW-Fi]l6%F^`nF^`nF^`n?(FiqFjqF[rF_`nF
]r?(FhxFjqF[rF_`nF]r?(F;FjqF[rF_`nF]r>&FW6%FiqFhxF;-Far6$*(,&FiqF[rFhxFd^lF[r,(
)F;F?FO)FhxF?Fd^l),&F;FOFhxFd^lF?Fd^lF[rF?Fd^lF?OFWF8F8F8F8,
I.calc3cocInvarf*6%F]xFgr'FB%'posintG67F_\mF`\mFivFjvF[wF\wF]wF^wF_wF`wFawFbwFc
wFdwFewFa\mFhwFiwFjwF[xF\xF8F8C+>FiqFjq>FgblF`jn@%29#""%>Fhy-%,co3SolutionGFay>
Fhy&9"6#Fcco>F_cl-FjyF\]l>Ff^l,&-F]\l6#-F`\l6#-Fc\l6$f*Ff\lF8Fg\lF8Fj\lF8F8F8FA
F[rF[rF[r?(F;F[rF[r,&-Fa]l6#FAF[rF[rF[rF]rC$>Fe^mFf^m>FacnFbcn>FbuFj^m?(Ff]lFjq
F[rFbalF]rC'?(FWF[rF[rFf^lF]r>F__mF`_m?(Fg_mF[rF[rFhdoF]r@%2Fjq&FAF[`m?(FioF[rF
[rFf^lF]rC&@$/Fio-F[]l6#Feeo>Ff`mFj`m@$/Fio,&FjeoF[rF[rF[r>Ff`m&F?Fcam@$2FioFje
o>Ff`mFfam@$2F_foFio>Ff`mFfam?(FioF[rF[rFf^lF]rC&@$Fieo>Ff`mF[dm@$F^fo>Ff`mFdam
@$Fcfo>Ff`mFfam@$Fffo>Ff`mFfam>Fi^l-Fibl6#,&FbtF[r&F[s6#FgdoFd^l>Fj[lF[em@$Fbem
?(FazFjqF[rFcfnF]rC'>FhxFjq?(F\zF[rF[rFgdoF]r>FhfnFaz?(F_^lF[rF[rFhdoF]r?(F]dlF
jqF[r,&-F[]l6#&FAFagnF[rFd^lF[rF]r>Fcgn&F?Fhgn?(F_pF[rF[rFhdoF]r@%2Fjq&FAF_u>Fh
x,&FhxF[r&Fhy6%&Ffhn6#,&-F[]l6#FjhoF[rFd^lF[r&F_fm6#Fbio&F_fm6#,&FbioF[rF[rF[rF
[r>Fhx,&FhxF[r&Fhy6%F_io&F[gmFeio&F[gmFgioFd^l>Fiq,&FiqF[r)F^hl-Fc\l6$f*6#F^hmF
8Fg\lF8-Far6$F?T$F8F86$FBFLFhxF[rOFiqF8F8F8F8,
I.CalcDihinvarsf*6%F%F&F-6)F/F0F1F4%*McocyclesG%&INVARGF'F8F8C->F_p,&FdhlF[rF[r
F[r>Fg_m-F`aoFido>F;-F=6/F?%'Dih_Z_GFAFBFCFDFE-FG6$-FJF_uFMFNFZFPFTFU>F;FinFco>
FWFjo-Fcp6$FWFhp-Fcp6(FWQhn3-cocycle~formula~f(x,y,z)=(x-y)[(2z^%d~-j^%d)-(2z-y
)^%d]/%d|+F8FAFAFAFA-Fcp6%FWFaq-F=6)F@FiqFdqFeqFEFj[pFN?(FiqFjqF[rFLF]rC$>Fio-F
gbo6&-F`y6$F_pFAF_sFAFg_m-Fcp6'FWQ+%s|*%s|*%s|+|+F8FdsFhs-FG6$FioFM-F\uFgoF8F8F
8F8