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o6$FigoF^o@$09.F^rC1>F]y-Fdcl6,F_hoFcelFioFifoFjfoQ#)_F=FgjlFfdlF]]mFdel>F\v-Fd
cl6.F_hoFielFioFifoFjfoFdhoFaoFfdlF]uFfdlFejlFdel>F]y-Fdgo6$FigoF]y>F\v-Fdgo6$F
igoF\v>Ffel-F]fl6$F]yF_fl>F[flF\fl-Fefl6$FgflF]yFdfl-F[gl6%FfelQgn#Mochizuki~2-
cocycle~solutions~for~the~Alexander~Quandle~%s|+F=-Fdcl6)FiamFioQ+[t^1,t^-1]F=Q
"/F=Q"(F=FjfoQ")F=-F[gl6%F[flQgn#Mochizuki~3-cocycle~solutions~for~the~Alexande
r~Quandle~%s|+F=Fgio-F[gl6'FfelQN#2-cocycle~formula~f(x,y)=(x-y)^%d^%d~*y^%d~|+
F=FioFgjlF]]m-F[gl6)F[flQfn#3-cocycle~formula~f(x,y,z)=(x-y)^%d^%d~*(y-z)^%d^%d
~*z^%d|+F=FioFaoFioF]uFejl-F[gl6%FfelF]hlF`dlF[hl>F_yFeo>FbhlFchl>FfhlFghlFihlF
[il-F[gl6$FbhlQ^o#TangleBraid~Mochizuki~2-cocycle~invariants~for~Alexander~Quan
dles|+F=-F[gl6$FfhlQ^o#TangleBraid~Mochizuki~3-cocycle~invariants~for~Alexander
~Quandles|+F=-F[gl6'FbhlFbjoFioFgjlF]]m-F[gl6)FfhlFejoFioFaoFioF]uFejl-F[gl6%Fb
hlQ7#Alexander~Quandle~%s|+F=Fgio-F[gl6%FfhlFg[pFgioF[jlF]jl?(FcoF`oFdoF^elFeoC
(>FjwF\dl>F\]l-Fdjn6+F_qF_qFioFaoF]uFejlFgjlF]]mF[[mFb[mFf\m@$09-F^rC0>Fep-Fdcl
6(Fb\pQ4TangleBraidColorsZ_F=FioFifoFjfoF\jo>Fep-Fdgo6$FigoFep>FepFg]n>Fi]mFj]m
-F[gl6%Fi]mQ_p#The~valid~color~vectors~for~the~TangleBraids~colored~with~the~Al
exander~quandle~%s|+F=Fgio-F[gl6%Fi]mF]hl&FjwF.>Fa_m-Fdcl6%Fi^mFh[m-Fdcl6(FiamF
ioQ$modF=F[joFjfoF\jo-F[gl6&Fi]mF\_mFa_mF]_m>Fafl,&-F_v6#-Fbv6#-Fev6$f*FhvF=Fiv
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Fc^o?(Fe_oFdoFdoFc^oFeo?(FdanFdoFdo,&F_dpFdoFdoFdoFeo>Fi_oFhy>F]y-FT6&F\vF_dpF_
yFc^o?(F_wFdoFdoFc^oFeoC%>FipF`o?(F\]lFdoFdoF_dpFeo@%2F`o&FejlF`]l>Fip,&FipFdo&
F^z6$&F^co6#-F]w6#Fcep&F^co6#,&FjepFdoFdoFdoFdo>Fip,&FipFdo&F^z6$&FhcoFiep&Fhco
F]fpFbx>FcoF^eoO6&FcoF]yF^zFc^oC$FgeoOF^rF=F=F=F=,
I;TangleBraidCalc2Mochinvarst+6*F_alFjqF^foF_foF`foFbal>FhalF^r>FjalF^rFcfoF=F=
C3>FgoF\dlF`dl>Fip-Fdcl6,FcpFddlFioFifoFjfoF\goFaoFfdlF]uF]go>FipFcgo@$F\]mC*>F
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felFbjoFioFaoF]uFfjo>F_yFeo>FbhlFchlFihlF[[p-F[gl6'FbhlFbjoFioFaoF]uFe[pF[jl?(F
coF`oFdoFejlFeoC'>FjwF\dl>F\]l-F^bp6(F_qF_qFioFaoF]uF[[mFb[m@$F]\nC0>Fep-Fdcl6(
FgjlFg\pFioFifoFjfoF\jo>FepFi\p>FepFg]n>Fi]mFj]mF]]pF`]p>Fa_mFd]p-F[gl6&Fi]mF\_
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oF\joF/Fg_pFaoFgclF]u>FbclF]`p?(FhtF`oFdoFgbnFeo?(F^zF`oFdoFgbnFeo-F[gl6(FfelF_
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F[jlFcdmF=F=F=6#F__mF=,
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F_yFft>F_uF^r>FjzFft>FjwFf[o>FhtF\\o>FjzF^\o@$F[]o>F_uFc]o@%F_uC.>F\v,&-F_v6#-F
bv6#-Fev6$f*FhvF=FivF=F\wF=F=F=FgjlFdoFdoFdo>Fc^oF`o>FayFe^o>Fgz-Fcw6&;F`o,&-F`
x6#FgjlFdoFbxFdoFgjlF*Ffw>F_yFdx>Fc^o-F"6(FayFhtFgzF\vFe]qF_y>F\yFc^o?(Fe_oFdoF
doFc^oFeo?(FdanFdoFdo,&Fe]qFdoFdoFdoFeo>Fi_oFhy?(FepFdoFdoF^^qFeo>F_`oF_z>F]y-F
T6&F\vFe]qF_yFc^o?(F_wFdoFdoFc^oFeo?(Ff[lF`oFdoF[[lFeoC'>F^oF`o?(Fh[lFdoFdoF^^q
Feo>F[aoFf[l?(FetFdoFdoFe]qFeo?(FawF`oFdo,&-F]w6#&FgjlFj`mFdoFbxFdoFeo>FdaoFhao
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o&F^co6#F]`q&F^co6#,&F]`qFdoFdoFdoFdo>F^o,&F^oFdo&Fjz6%Fj_q&FhcoF``q&FhcoFb`qFb
x>FgoF[doO6&FgoF]yFjzFc^oC$FgeoOF^rF=F=F=F=,
I;TangleBraidCalc3Mochinvarst+6+F_alFjqF^foFijnFjjnF[[oFbal>FhalF^r>FjalF^rFcfo
F=F=C3>FgoF\dlF`dl>F^oF_go>F^oF[ho@$F]elC*>F\v-Fdcl6.F^elFielFioFifoFjfoFdhoFao
FfdlF]uFfdlFejlFdel>F\vF\io>F[flF\flFdflF]joFcjoF[hl>F_yFeo>FfhlFghlF[ilF^[pFc[
pFh[pF]jl?(FcoF`oFdoFgjlFeoC'>FjwF\dl>F\]l-Ff[q6)F_qF_qFioFaoF]uFejlF[[mFahn@$F
\]mC0>Fep-Fdcl6(F]]mFg\pFioFifoFjfoF\jo>FepFi\p>FepFg]n>Fi]mFj]mF]]pF`]p>Fa_mFd
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mFeo?(FetFdoFdoF\`mFeo?(Fh[lFdoFdoFaflFeoF_jp>FjwF\dlF`]pF\amF_am@$F^_pC+>Ff[lF
i_p>FbclF]`p?(FhtF`oFdoFgbnFeo?(F^zF`oFdoFgbnFeo?(FjzF`oFdoFgbnFeo-F[gl6)F[flFe
bmFf[lFhtF^zFjz-Fi[m6$&Ff`mFj`pF]\m>FjwF\dlFb]pFdapFecmFicm>F_yF^rF^dm>FgoF\dlF
`dlF]jlFedmF=F=F=6#""(F=,
I'ismultf*6#%)charstarGF=F=F=@%/FcpQ"*F=OFeoOF^rF=F=F=F=,
I9TangleBraidCalcDihinvarst+6'F_alFjq%.numberofKnotsG>FhalF^r>FjalF^r69FfnF\blF
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jblF^clF_clF=F=C4>FepFft>Ffhl,&FbdoFdoFdoFdo>Fep-%6MochizukiDihedral3cocG6#Fio>
FgoF\dlF`dl>Fip-Fdcl6(FcpQ6TangleBraidMochDih_R_F=FioQ*3coc_inZ_F=FioF]go@$0F]u
F^rC(>Fi]m-Fdcl6&F]uQ4MochDih3cocSOL_inZ_F=FioFdel>F]y-F]fl6$Fi]mF_fl-F[gl6%F]y
Qin#Mochizuki~3-cocycle~solutions~for~the~Dihedral~Quandle~R_%d~|+F=Fio-F[gl6*F
]yQ[p#3-cocycle~formula~in~Z_%d~is~f(x,y,z)=(x-y)[(2z^%d~-j^%d)-(2z-y)^%d]/%d~m
od~%d|+F=FioFioFioFioFioFio-Fefl6$FgflFi]m>FetFeoFihl>F^oFchl-F[gl6%F^oQfo#Tang
leBraid~Mochizuki~3-cocycle~invariants~for~the~Dihedral~Quandle~R_%d~|+F=Fio-F[
gl6*F^oF]hqFioFioFioFioFioFio-F[gl6%F^oF]hlF`dl?(FcoF`oFdoF_qFeoC(>Fh[lF\dl&Fh[
lF.>Fbhl-F_`n6&-Fg[o6$FfhlFioFioF[[mFep-F[gl6'F^oQ,%s|*%s|*|+%s|+|+F=Fe[mF^\m-F
i[m6$&FbhlFe\mF]\m@$0FaoF^rC/>F\]l-Fdcl6%FaoQ7TangleBraidColorDih_R_F=Fio>F\]l-
Fdcl6&F\]lFi]nF[\mFg]m>Fbcl-F]fl6$F\]lF_fl-F[gl6%FbclQgo#The~valid~color~vector
s~for~the~TangleBraids~colored~with~the~quandle~R_%d|+F=Fio-F[gl6%FbclF]hlF\iq>
Fafl-Fdcl6'Fi^mFh[m%"RGFgclFio-F[gl6&FbclF\_mFafl&FbhlF3>Fa_m,&-F_v6#-Fbv6#-Fev
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gl6$FbclF^am-F`amFjcl@$0FetF^rC*>Ff[l-Fdcl6%%-co3MochDihZ_GFio%$SolG>F`[lFio?(F
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a~finite~ring.F=OF=%)continueG>FbclFit>Fbhl-F`p6%FfbnFfbn;F`o,&Fa_mFdoFbxFdo>Fe
pFft?(F^oF`oFdoFgbnFeo?(F[flF`oFdoFgbnFeo>&Ffhl6$F^oF[fl-%$sumG6$*&&Fbhl6%F^oF[
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eoC&>&Ff[l6%FcoFgoFip,,*&FbdoFdo&FfhlFjemFdoFdo&Ffhl6$&FcpFjemFipFdo*&FbdoFdo&F
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*genmatrixG6$FepFi]m>F_y-%'vectorG6$-&Fbhr6#%'rowdimGF`]lF`o>F\v-Fd_l6$-%)Linso
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edSol~solutions~are~not~validF=OF=F=F=F=F=,
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FifmF[cm>F\]l-Fh^r6%FcpF_qFao>F\]l&FSFjbm?(FcoF`oFdo,&FfhlFdoFbxFdoFeo?(FgoF`oF
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cw6&;F`o,&-F`xFgfqFdoFbxFdoFioF*Ffw>F_wFdx>Fgz-F"6(Fe_oFfhlFdanF]yFfasF_w>&F`[l
F_oFgz?(F^oFdoFdoFgzFeo?(FepFdoFdo,&FfasFdoFdoFdoFeo>&&F`[lFeqF`q-F`p6$;FdoF]yF
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F=F=,
I<TangleBraidCalc2twistinvarst+6*F_alFjqF^foF`alFaalFbal>FhalF^r>FjalF^rF[blF=F
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\flFdfl-F[gl6(F[flQdq#Twisted~Quandle~2-cocycle~solutions~for~the~Quandle~Q%d_%
d~and~|+#coefficient~group~Z_%d[t,t^(-1)]/(%s)|+|+F=FaoF]uFioFjfo-F[gl6&F[flFcg
lFaoF]uFfglF[hl>FayFeo>FbhlFchlFihl-F[gl6(FbhlQar#TangleBraid~Twisted~Quandle~2
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1)]/(%s)|+|+F=FaoF]uFioFjfo-F[gl6&FbhlFcglFaoF]uFgilF[jl?(FcoF`oFdoFejlFeoC'>F_
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Fi^mFh[mFgclFfclFaoFgclF]uF^^n>Fa_m,&-F_v6#-Fbv6#-Fev6$f*FhvF=FivF=F\wF=F=F=F[[
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