############################################### #The following code calculates the three overlaps #r_6, r_7, r_8 on page 8 of the manuscript. ############################################### r1_sub_list:= {seq(seq(x[r,s,2,1]= p*x[r,s,1,2] + m*x[r,s,1,1], r = 1..3), s = 1..3)} union {seq(seq(x[r,2,1,s]= p*x[r,1,2,s]+ m*x[r,1,1,s], r = 1..3), s = 1..3)} union {seq(seq(x[2,1,r,s]= p*x[1,2,r,s]+ m*x[1,1,r,s], r = 1..3), s = 1..3)}; r2_sub_list:={seq(seq(x[r,s,3,2] = a*x[r,s,3,1] + n*x[r,s,1,3] + b*x[r,s,2,3], r =1..3), s = 1..3)} union {seq(seq(x[r,3,2,s] = a*x[r,3,1,s] + n*x[r,1,3,s] + b*x[r,2,3,s], r = 1..3), s = 1..3)} union {seq(seq(x[3,2,r,s] = a*x[3,1,r,s] + n*x[1,3,r,s] + b*x[2,3,r,s], r = 1..3), s = 1..3)}; r3_sub_list:={seq(x[r,3,3,1] = c*x[r,1,3,3] + d*x[r,2,3,3]+e*x[r,3,1,3], r = 1..3)} union {seq(x[3,3,1,r] = c*x[1,3,3,r] +d*x[2,3,3,r]+ e*x[3,1,3,r], r = 1..3)}; r4_sub_list:= {seq(x[r,3,1,1] = f*x[r,1,1,3] + g*x[r,1,2,3] +h*x[r,2,2,3] + j*x[r,1,3,1] + k *x[r,2,3,1], r = 1..3)} union {seq(x[3,1,1,r] = f*x[1,1,3,r] + g*x[1,2,3,r] + h*x[2,2,3,r] + j*x[1,3,1,r] + k *x[2,3,1,r], r = 1..3)}; r5_sub_list:= {seq(x[r,3,1,2] = q*n*x[r,1,3,1] + q*b*x[r,2,3,1] + z*x[r,3,1,1], r = 1..3)} union {seq(x[3,1,2,r] = q*n*x[1,3,1,r] + q*b*x[2,3,1,r] + z*x[3,1,1,r], r = 1..3)}; way1:= subs(r3_sub_list, x[3,3,1,2]); way2:= subs(r5_sub_list, x[3,3,1,2]); resolve11:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way1)))))))); resolve12:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve11)))))))); resolve13:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve12)))))))); resolve14:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve13)))))))); resolve15:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve14)))))))); resolve1:=subs(r4_sub_list,resolve15); resolve21:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way2)))))))); resolve22:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve21)))))))); resolve23:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve22)))))))); resolve24:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve23)))))))); resolve25:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve24)))))))); resolve2:=subs(r4_sub_list,resolve25); AA:=simplify(resolve1 - resolve2); BB:={seq(seq(seq(seq(coeff(AA, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; R6:=add(add(add(add(coeff(AA, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s = 1..3), t = 1..3), u = 1..3); way3:= subs(r3_sub_list, x[3,3,1,1]); way4:= subs(r4_sub_list, x[3,3,1,1]); resolve31:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way3)))))))); resolve32:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve31)))))))); resolve33:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve32)))))))); resolve34:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve33)))))))); resolve3:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve34)))))))); resolve41:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way4)))))))); resolve42:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve41)))))))); resolve43:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve42)))))))); resolve44:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve43)))))))); resolve4:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve44)))))))); AAA:=simplify(resolve3 - resolve4); BBB:={seq(seq(seq(seq(coeff(AAA, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; R7:=add(add(add(add(coeff(AAA, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s = 1..3), t = 1..3), u = 1..3); way5:= subs(r1_sub_list, x[3,1,2,1]); way6:= subs(r5_sub_list, x[3,1,2,1]); resolve51:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way5)))))))); resolve52:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve51)))))))); resolve53:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve52)))))))); resolve54:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve53)))))))); resolve5:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve54)))))))); resolve61:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way6)))))))); resolve62:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve61)))))))); resolve63:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve62)))))))); resolve64:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve63)))))))); resolve6:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve64)))))))); AAAA:=simplify(resolve5 - resolve6); BBBB:={seq(seq(seq(seq(coeff(AAAA, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; R8:=add(add(add(add(coeff(AAAA, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s = 1..3), t = 1..3), u = 1..3); ######################################################## Output from the above program: ###################################################### R6; (-q*b*a-q*n+e*a-z*e)*x[3,1,3,1]+(e*n*j+c*a*e+c*b*a+c*n-z*c*e+e*b*z*j)*x[1,3,1,3] +(e*n*k+d*b*a+d*a*e+d*n-z*d*e+e*b*z*k)*x[2,3,1,3] +(e*n*f+c*b*n+a*c^2-z*c^2+d*a*c*m+d*b*n*m-q*b*n*c-q*b^2*c*m+e*b*z*f-z*d*c*m)*x[1,1,3,3] +(e*n*g+c*a*d+c*b^2+d*a*c*p+d*b*n*p-q*b*n*d-q*b^2*c*p-z*c*d+e*b*z*g-z*d*c*p)*x[1,2,3,3] +(e*n*h-q*b^2*d-z*d^2+a*d^2+d*b^2+e*b*z*h)*x[2,2,3,3] R7; (-k*a+e-j)*x[3,1,3,1]+(-f*j+c*e-g*z*j-g*q*n-h*n*a-e*n*k-h*a*q*n-h*a*z*j)*x[1,3,1,3] +(d*e-e*b*k-g*z*k-g*q*b-f*k-h*b*a-h*a*q*b-h*a*z*k)*x[2,3,1,3] +(-h*n^2+d*c*m+c^2-g*z*f-f^2-k*n*c-h*a*z*f-h*b*n*m-k*b*c*m)*x[1,1,3,3] +(d*c*p+c*d-f*g-z*g^2-h*n*b-k*n*d-h*a*z*g-h*b*n*p-k*b*c*p)*x[1,2,3,3] +(-h*b^2+d^2-g*z*h-f*h-a*z*h^2-k*b*d)*x[2,2,3,3] R8; (-z*f+m*j^2+m*f+p*z*j^2+p*f*a-j*q*n-z*j^2-k*z*j*m+p*j*q*n+k*j*m^2+p*k*q*n*m+p*k*z*j*m-q*b*j*m)*x[1,1,3,1] +(-z*g+m*g+p*g*a-j*z*k+m*j*k-q*n*k+k*q*n*p^2+k*z*j*p^2+m*k*j*p)*x[1,2,3,1] +(m*h+m*k^2-z*h+p*h*a-k*q*b-z*k^2+p*k*q*b+p*z*k^2)*x[2,2,3,1] +(p*f*n-j*z*f+h*n*p*m^2-k*z*f*m+p*g*n*m+p*j*z*f+m*j*f+k*f*m^2-q*n*f+h*n*p^2*m^2+k*z*f*p^2*m+k*f*p*m^2-q*b*f*p*m-q*b*f*m)*x[1,1,1,3] +(g*n*p^2+p*f*b-g*z*j+h*n*m*p^2-k*z*f*p^2-k*z*g*m+p*j*z*g+m*j*g+k*g*m^2-g*q*n +h*n*m*p^3+k*z*f*p^3+p*k*z*g*m+m*k*f*p^2-q*b*f*p^2-q*b*g*m)*x[1,1,2,3] +(h*n*p^3+p*g*b-j*z*h-k*z*g*p+p*j*z*h+m*j*h-q*n*h+k*z*g*p^2+m*k*g*p-q*b*g*p)*x[1,2,2,3] +(p*h*b-k*z*h+p*k*z*h+m*k*h-q*b*h)*x[2,2,2,3] ############################################################### # In section 3.3, we show that in the case m = 0, p \neq 1, #further reductions allow us to assume that n = 1, #d not equal to bc. # We can also assume b = 0 or b =1, since if b is nonzero #the example is twist equivalent to an example with b = 1. # In section 3.4, we show that in case m=1, p =1, further #reductions allow us to assume that a = 0 and that we # do not have c = d= 0. # The following commands find these solutions #(recall that also pq =1 and z = qa -qm.) ################################################################## Sb0:=solve(BB union BBB union BBBB union {q*p-1} union {d <> b*c} union {m = 0} union {p <>1} union {n = 1} union {b =0} union {z=a*q-q*m},{a,b,c,d,e,q,p,f,g,h,j,k,m,n,z}); Sb1:=solve(BB union BBB union BBBB union {q*p-1} union {d <> b*c} union {m = 0} union {p <>1} union {n = 1} union {b =1} union {z=a*q-q*m},{a,b,c,d,e,q,p,f,g,h,j,k,m,n,z}); Sc:=solve(BB union BBB union BBBB union {q = 1} union {m= 1} union {p = 1} union {a=0} union {z=a*q-q*m} union {c <> 0},{a,b,c,d,e,q,p,f,g,h,j,k,m,n,z}); Sd:=solve(BB union BBB union BBBB union {q = 1} union {m= 1} union {p = 1} union {a=0} union {z=a*q-q*m} union {d <> 0},{a,b,c,d,e,q,p,f,g,h,j,k,m,n,z}); ################################# #Output from these commands: ################################## Sb0; (no solutions) Sb1; {a = -1/(q^2), b = 1, c = -q^3, d = 0, e = q^2+q, f = -q^3, g = 0, h = 0, j = q^2+q, k = 0, m = 0, n = 1, p = 1/q, q = q, z = -1/q}, {a = -1, b = 1, c = 1, d = 0, e = 0, f = 0, g = -1, h = 0, j = 1, k = 1, m = 0, n = 1, p = -1, q = -1, z = 1}, {a = -1, b = 1, c = 1, d = 0, e = 0, f = 0, g = 1, h = 0, j = 1, k = 1, m = 0, n = 1, p = -1, q = -1, z = 1}, {a = -1, b = 1, c = 1, d = 0, e = 0, f = h-1, g = 0, h = h, j = 0, k = 0, m = 0, n = 1, p = -1, q = -1, z = 1}, {a = -1, b = 1, c = 0, d = 1, e = 0, f = 0, g = RootOf(_Z^2+1), h = 0, j = 1, k = 1, m = 0, n = 1, p = -1, q = -1, z = 1}, {a = -RootOf(_Z^2+_Z+1), b = 1, c = -1, d = 0, e = -1, f = -1, g = 0, h = 0, j = -1, k = 0, m = 0, n = 1, p = -RootOf(_Z^2+_Z+1)-1, q = RootOf(_Z^2+_Z+1), z = 1+RootOf(_Z^2+_Z+1)}, {a = 1+RootOf(_Z^2+_Z+1), b = 1, c = RootOf(_Z^2+_Z+1), d = -1, e = -1, f = 0, g = 0, h = -1, j = RootOf(_Z^2+_Z+1), k = -1, m = 0, n = 1, p = RootOf(_Z^2+_Z+1), q = -RootOf(_Z^2+_Z+1)-1, z = -RootOf(_Z^2+_Z+1)}, {a = -1, b = 1, c = 0, d = 1, e = 0, f = 1-RootOf(1+2*_Z^2-2*_Z), g = 0, h = RootOf(1+2*_Z^2-2*_Z), j = 0, k = 0, m = 0, n = 1, p = -1, q = -1, z = 1}, {a = RootOf(_Z^2-_Z+1), b = 1, c = 0, d = RootOf(_Z^2-_Z+1)-1, e = -1, f = 0, g = -RootOf(_Z^2-_Z+1), h = 0, j = 0, k = RootOf(_Z^2-_Z+1)-1, m = 0, n = 1, p = RootOf(_Z^2-_Z+1)-1, q = -RootOf(_Z^2-_Z+1), z = 1-RootOf(_Z^2-_Z+1)} Sbc; {a = 0, b = 1/2*e, c = -1/4*e^2, d = 0, e = e, f = -1/4*e^2, g = 0, h = 0, j = e, k = 0, m = 1, n = e, p = 1, q = 1, z = -1} Sbd; (no solutions) ############################################################ #We see that Sb0 is empty (so there are no solutions #with b = 0) and we find nine solution families for section #3.3 in the list Sb1, #although one turns out to be a repeat. These have been #rearranged above so that in the notation of the paper they #correspond to A(1,q), #B(1), C(1), D(1,h), E(1, \gamma), F(1, \gamma), #\underline{F}(1, \gamma), G(1, \gamma), and the #last one is just a repeat of F(1, \gamma) #written in a different way. # #We see examining Sbc and Sbd that there is #one solution in Section 3.4, as stated in the paper. ######################################################## ############################################################ #The following code repeats the process above with a new #r_4 such that l = 0 and k = -1, as discussed in section 3.5. ####################################################### r1_sub_list:= {seq(seq(x[r,s,2,1]= p*x[r,s,1,2] + m*x[r,s,1,1], r = 1..3), s = 1..3)} union {seq(seq(x[r,2,1,s]= p*x[r,1,2,s]+ m*x[r,1,1,s], r = 1..3), s = 1..3)} union {seq(seq(x[2,1,r,s]= p*x[1,2,r,s]+ m*x[1,1,r,s], r = 1..3), s = 1..3)}; r2_sub_list:={seq(seq(x[r,s,3,2] = a*x[r,s,3,1] + n*x[r,s,1,3] + b*x[r,s,2,3], r =1..3), s = 1..3)} union {seq(seq(x[r,3,2,s] = a*x[r,3,1,s] + n*x[r,1,3,s] + b*x[r,2,3,s], r = 1..3), s = 1..3)} union {seq(seq(x[3,2,r,s] = a*x[3,1,r,s] + n*x[1,3,r,s] + b*x[2,3,r,s], r = 1..3), s = 1..3)}; r3_sub_list:={seq(x[r,3,3,1] = c*x[r,1,3,3] + d*x[r,2,3,3]+e*x[r,3,1,3], r = 1..3)} union {seq(x[3,3,1,r] = c*x[1,3,3,r] +d*x[2,3,3,r]+ e*x[3,1,3,r], r = 1..3)}; r4_sub_list:= {seq(x[r,2,3,1] = f*x[r,1,1,3] + g*x[r,1,2,3] +h*x[r,2,2,3] + j*x[r,1,3,1], r = 1..3)} union {seq(x[2,3,1,r] = f*x[1,1,3,r] + g*x[1,2,3,r] + h*x[2,2,3,r] + j*x[1,3,1,r], r = 1..3)}; r5_sub_list:= {seq(x[r,3,1,2] = q*n*x[r,1,3,1] + q*b*x[r,2,3,1] + z*x[r,3,1,1], r = 1..3)} union {seq(x[3,1,2,r] = q*n*x[1,3,1,r] + q*b*x[2,3,1,r] + z*x[3,1,1,r], r = 1..3)}; way1:= subs(r3_sub_list, x[3,3,1,2]); way2:= subs(r5_sub_list, x[3,3,1,2]); resolve11:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way1)))))))); resolve12:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve11)))))))); resolve13:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve12)))))))); resolve14:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve13)))))))); resolve15:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve14)))))))); resolve1:=subs(r4_sub_list,resolve15); resolve21:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way2)))))))); resolve22:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve21)))))))); resolve23:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve22)))))))); resolve24:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve23)))))))); resolve25:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve24)))))))); resolve2:=subs(r4_sub_list,resolve25); AAk:=simplify(resolve1 - resolve2); BBk:={seq(seq(seq(seq(coeff(AAk, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; rr6:=add(add(add(add(coeff(AAk, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s = 1..3), t = 1..3), u = 1..3); way3:= subs(r2_sub_list, x[3,2,3,1]); way4:= subs(r4_sub_list, x[3,2,3,1]); resolve31:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way3)))))))); resolve32:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve31)))))))); resolve33:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve32)))))))); resolve34:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve33)))))))); resolve3:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve34)))))))); resolve41:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way4)))))))); resolve42:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve41)))))))); resolve43:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve42)))))))); resolve44:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve43)))))))); resolve4:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve44)))))))); AAAk:=simplify(resolve3 - resolve4); BBBk:={seq(seq(seq(seq(coeff(AAAk, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; rr7:=add(add(add(add(coeff(AAAk, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s= 1..3), t = 1..3), u = 1..3); way5:= subs(r1_sub_list, x[3,1,2,1]); way6:= subs(r5_sub_list, x[3,1,2,1]); resolve51:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way5)))))))); resolve52:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve51)))))))); resolve53:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve52)))))))); resolve54:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve53)))))))); resolve5:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve54)))))))); resolve61:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way6)))))))); resolve62:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve61)))))))); resolve63:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve62)))))))); resolve64:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve63)))))))); resolve6:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve64)))))))); AAAAk:=simplify(resolve5 - resolve6); BBBBk:={seq(seq(seq(seq(coeff(AAAAk, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; rr8:=add(add(add(add(coeff(AAAAk, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s = 1..3), t = 1..3), u = 1..3); way7:= subs(r4_sub_list, x[2,3,1,2]); way8:= subs(r5_sub_list, x[2,3,1,2]); resolve71:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way7)))))))); resolve72:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve71)))))))); resolve73:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve72)))))))); resolve74:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve73)))))))); resolve7:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve74)))))))); resolve81:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, way8)))))))); resolve82:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve81)))))))); resolve83:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve82)))))))); resolve84:=subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r4_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve83)))))))); resolve8:=subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list, subs(r1_sub_list, subs(r5_sub_list, subs(r3_sub_list, subs(r2_sub_list,subs(r1_sub_list, resolve84)))))))); AAAAAk:=simplify(resolve7 - resolve8); BBBBBk:={seq(seq(seq(seq(coeff(AAAAAk, x[r,s,t,u]), r = 1..3), s = 1..3), t = 1..3), u = 1..3)}; rr9:=add(add(add(add(coeff(AAAAAk, x[r,s,t,u])*x[r,s,t,u], r = 1..3), s = 1..3), t = 1..3), u = 1..3); ######################### #output from the above: ######################### rr6; (-q*b*a-q*n+e*a-z*e)*x[3,1,3,1]+(e*n+e*b*z)*x[3,1,1,3] +(d*a*e*j+d*b*a*j-z*d*e*j+c*a*e+c*b*a+d*n*j+c*n-z*c*e)*x[1,3,1,3] +(d*b*a*f+d*a*e*f-z*d*e*f+c*b*n+d*n*f+a*c^2-z*c^2+d*a*c*m+d*b*n*m-q*b*n*c-q*b^2*c*m-z*d*c*m)*x[1,1,3,3] +(d*a*e*g+d*b*a*g-z*d*e*g+c*a*d+d*n*g+c*b^2+d*a*c*p+d*b*n*p-q*b*n*d-q*b^2*c*p-z*c*d-z*d*c*p)*x[1,2,3,3] +(d*a*e*h+d*b*a*h-q*b^2*d-z*d*e*h+d*n*h-z*d^2+a*d^2+d*b^2)*x[2,2,3,3] rr7; (a-j)*x[3,1,3,1] +(-f-z*g-h*a*z)*x[3,1,1,3] +(e*n-h*a*q*b*j+b*e*j-g*q*n-h*n*a-h*a*q*n-g*q*b*j-h*b*a*j)*x[1,3,1,3] +(-h*n^2+c*n-h*a*q*b*f+b*e*f+b*c*m-h*b*n*m-g*q*b*f-h*b*a*f)*x[1,1,3,3] +(d*n-h*a*q*b*g+b*e*g+b*c*p-h*n*b-h*b*n*p-q*b*g^2-h*b*a*g)*x[1,2,3,3] +(b*d-h*b^2+b*e*h-b*a*h^2-g*q*b*h-a*q*b*h^2)*x[2,2,3,3] rr8; (m-z)*x[3,1,1,1] +(-q*n-q*b*j)*x[1,3,1,1] +(-q*b*f-g*q*b*j-q*b*h*p*j^2-q*b*h*j*m)*x[1,1,3,1] +p*x[3,1,1,2]+(-g*q*b*f-q*b*h*f*p*m-q*b*h*f*m-q*b*h*j*p*f)*x[1,1,1,3] +(-q*b*g^2-q*b*h*f*p^2-q*b*h*g*m-q*b*h*j*p*g)*x[1,1,2,3] +(-g*q*b*h-q*b*h*g*p-q*b*j*p*h^2)*x[1,2,2,3]-q*b*h^2*x[2,2,2,3] rr9; (f*a+g*a*j+j*q*n+q*b*j^2-q*n*m-z*g*j-z*f+h*a*p*j^2+h*a*j*m-q*n*p*j-q*b*p*j^2-q*b*j*m-z*h*p*j^2-z*h*j*m)*x[1,1,3,1] +(f*n+g*a*f+g*n*m+h*n*m^2-z*g*f+h*a*f*p*m+h*a*f*m+h*a*j*p*f+h*n*p*m^2 +j*q*b*f-q*n*p*f-q*b*f*p*m-q*b*f*m-q*b*j*p*f-z*h*f*p*m-z*h*f*m-z*h*j*p*f)*x[1,1,1,3] +(a*g^2+f*b+g*n*p-z*g^2+h*a*f*p^2+h*a*g*m+h*a*j*p*g+h*n*p*m+h*n*m*p^2 +g*q*b*j-q*n*p*g-q*b*f*p^2-q*b*g*m-q*b*j*p*g-z*h*f*p^2-z*h*g*m-z*h*j*p*g)*x[1,1,2,3] +(g*b+g*a*h+h*n*p^2-z*g*h+h*a*g*p+a*j*p*h^2+j*q*b*h-q*n*p*h-q*b*g*p-q*b*j*p*h-z*h*g*p-z*j*p*h^2)*x[1,2,2,3] +(-q*b*h+a*h^2+h*b-z*h^2)*x[2,2,2,3] ###################### #As described in section 3.5 of the paper, we solve #for the coefficients of r_6, r_7, r_9 to be 0, #in the same cases #as before. ##################### Sb0k:=solve(BBk union BBBk union BBBBBk union {q*p-1} union {d <> b*c} union {m = 0} union {p <>1} union {n = 1} union {b =0} union {z=a*q-q*m},{a,b,c,d,e,q,p,f,g,h,j,m,n,z}); Sb1k:=solve(BBk union BBBk union BBBBBk union {q*p-1} union {d <> b*c} union {m = 0} union {p <>1} union {n = 1} union {b =1} union {z=a*q-q*m},{a,b,c,d,e,q,p,f,g,h,j,m,n,z}); Sck:=solve(BBk union BBBk union BBBBBk union {q = 1} union {m= 1} union {p = 1} union {a=0} union {z=a*q-q*m} union {c <> 0},{a,b,c,d,e,q,p,f,g,h,j,m,n,z}); Sdk:=solve(BBk union BBBk union BBBBBk union {q = 1} union {m= 1} union {p = 1} union {a=0} union {z=a*q-q*m} union {d <> 0},{a,b,c,d,e,q,p,f,g,h,j,m,n,z}); ################### #All outputs are empty--- no solutions. ################# ################################ #The final calculation necessary to Section 3.5 is to #assume now that r_4 has l = 0, k = 0, h \neq 0 and calculate #that the leading term of the relation arising from the #overlap x_3x_2^x_3 is -jx_3x_1x_3x_1. This is easily #done by hand and so #there is no need to revise the program to do this. ###########################