FG-monodromies-SL4.sagews

load('new-feed.sage')

R.<y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12,y13,y14,y15,y16,y17,y18>=QQ[]


### quiver is sl4-originalstyle in 2016-quivers

baseC=matrix(QQ,[[0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0],[-1,0,1,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0],[1,-1,0,-1,0,1,0,0,0,1,-1,0,0,0,0,0,0,0],[0,0,0,0,-1,0,1,0,0,0,1,0,-1,0,0,0,0,0],[0,1,-1,0,0,-1,0,1,0,0,0,0,1,-1,0,0,0,0],[0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0],[0,0,0,1,-1,0,0,0,-1,0,1,0,0,0,0,1,-1,0],[0,0,0,0,1,-1,0,0,0,-1,0,1,0,0,0,0,1,-1],[0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0],[0,0,0,0,0,1,-1,0,0,0,0,-1,0,1,0,0,-1,1],[0,0,0,0,0,0,1,-1,0,0,0,0,-1,0,1,-1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0],[0,0,0,0,0,0,0,0,1,-1,0,0,0,1,-1,0,0,0],[0,0,0,0,0,0,0,0,0,1,-1,0,1,-1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0]]);

baseC[0,3]=1/2
baseC[3,8]=1/2
baseC[8,3]=-1/2
baseC[3,0]=-1/2

baseC[14,7]=1/2
baseC[7,2]=1/2
baseC[2,7]=-1/2
baseC[7,14]=-1/2

def get_monomials(xpr):
    d=xpr.denominator()
    n=xpr.numerator()
    terms = [a*b for (a,b) in zip(n.monomials(),n.coefficients())]
    return [t/d for t in terms]
def how_long(x):
    return len(get_monomials(x))

def show_monomials(xpr):
    for m in get_monomials(xpr):
        print m
        
        
        
def sym(vs,deg):
    if (1 in vs) or (deg>len(vs)) or (deg<0):
        return False
    else:
        k = len(vs)-deg
        p = 1
        for v in vs:
            p = p*(y1+y(v))
        res = diff(p,y1,k)/factorial(k)
        return res.subs(y1=0)

def get_path(xpr):
    res=[]
    l=len(get_monomials(xpr))
    
    for ix in range(l-1):
        try:
            res.append( get_monomials(xpr)[ix]/get_monomials(xpr)[ix+1])
        except:
            ValueError
    res.append(get_monomials(xpr)[-1])
    res.reverse()
    return res


def build_feed(f,muts):
    res=copy(f)
    for m in muts:
        if type(m)!=type((0,1)):
            res.mutate(m)
        else:
            res.geom_mutate(m[0],m[1])
        res.vList=R.gens()
    return res

def PB(f,g,mat=baseC,ring=R):
    v=ring.gens()
    res=0
    l=len(v)
    for i in range(l):
	    for j in range(l):
		    res += diff(f,v[i])*diff(g,v[j])*mat[i,j]*v[i]*v[j]
    return res

def to_mon(vec):
    res=1
    for i,v in enumerate(vec):
        res=res*R.gens()[i]**v
    return res

e1 = build_path([1,2])
e2 = build_path([4,5,6,7])
e3 = build_path([9,10,11,12,13,14])

f1 = build_path([3,7,13,18,11,5])
f2 = build_path([8,14,17,10])
f3 = build_path([15,16])

### quiver is SL4

initial_feed = feed(baseC,R.gens())

initial_feed.inv_mutation_auto([5,11,18,13,7],f1)

y3

def H1(x):
    return matrix([[1,0,0,0],[0,x,0,0],[0,0,x,0],[0,0,0,x]])

def H2(x):
    return matrix([[1,0,0,0],[0,1,0,0],[0,0,x,0],[0,0,0,x]])
def H3(x):
    return matrix([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,x]])

F1 = matrix([[1,0,0,0],[1,1,0,0],[0,0,1,0],[0,0,0,1]])
E1=F1.transpose()
F2 = matrix([[1,0,0,0],[0,1,0,0],[0,1,1,0],[0,0,0,1]])
E2=F2.transpose()
F3 = matrix([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,1,1]])
E3=F3.transpose()

w = -matrix([[0,0,0,1],[0,0,-1,0],[0,1,0,0],[-1,0,0,0]])


def hexa(x1,x2,x3):
    return F3*F2*H3(x1)*F3*F1*H2(x2)*F2*H3(x3)*F3

def rect(z1,z2,z3):
    return H1(z1)*H2(z2)*H3(z3)

def mutate_matrix(feed,muts,M,ring=R):
    n=M.ncols()
    res=matrix(FractionField(R),n,n)
    for i in range(n):
        for j in range(n):
            res[i,j] = feed.inv_mutation_auto(muts,M[i,j])
    return res

###### First let's look at the un-self-folded triangulation:
###### Formula for E's:

upper_path = rect(y1,y4,y9)*hexa(y10,y5,y11)*rect(y2,y6,y12)*hexa(y13,y7,y14)*rect(y3,y8,y15)

lower_path = rect(y15,y8,y3)*hexa(y7,y14,y13)*rect(y16,y17,y18)*hexa(y11,y10,y5)*rect(y9,y4,y1)

s_upper_path = rect(y1,y4,y9)*hexa(y10,y5,y11)*rect(y2,y6,y12)*hexa(y13,y7,y14)
s_lower_path = rect(y16,y17,y18)*hexa(y11,y10,y5)*rect(y9,y4,y1)

lower_path[0,0]
lower_path[1,1]
lower_path[2,2]
lower_path[3,3]

1 y9*y15*y16 y4*y8*y9*y10*y14*y15*y16*y17 y1*y3*y4*y5*y7*y8*y9*y10*y11*y13*y14*y15*y16*y17*y18

#gsv_muts=[(12,18),6,2,11,5,13,7,10,6,14,17,16,11,13,11]
gsv_muts=[12,6,2,11,5,13,7,10,6,14,17,16,11,13,11]

XX=matrix([[0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,0,-1,-1,1,0,-1,1,0,0,0,0,0,0,0,0,0,0],[-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,-1,0,1,0,-1,1,0,-1,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,-1,0,0,0,-1,0,0,0,0,1,0,0],[0,1,0,0,-1,1,0,-1,0,0,0,0,0,-1,1,0,0,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0],[0,0,0,0,-1,0,0,0,1,0,0,0,0,-1,0,1,0,0],[0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,-2,1,1],[0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,1,0,0,0,0,-1,-1,0,0],[0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,-1,0,0,0,-1,2,-1,-1,1,0,0,-1,-1],[0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0]]);

gsv_feed=build_feed(initial_feed,gsv_muts)

gsv_mat = gsv_feed.exchangeMatrix

gsv_mat[0,3]=1/2
gsv_mat[3,8]=1/2
gsv_mat[8,3]=-1/2
gsv_mat[3,0]=-1/2

gsv_mat[14,7]=1/2
gsv_mat[7,2]=1/2
gsv_mat[2,7]=-1/2
gsv_mat[7,14]=-1/2

def is_central(xpr,mat=gsv_mat):
    for v in R.gens():
        if PB(v,xpr,mat)!=0:
            return False
    return True

def cf(i,j,mat=gsv_mat):
    return PB(i,j,mat)/(i*j)

U_init = upper_path*w*lower_path*w.inverse()
s_U_init = rect(y1,y4,y9)*hexa(y10,y5,y11)*rect(y2,y6,y12)*w*(hexa(y14,y13,y7)^-1) *rect(y18,y17,y16)^-1* (hexa(y5,y11,y10)^-1)*w*(hexa(y10,y5,y11)^-1)*rect(y1,y4,y9)^-1

U_init.matrix_from_rows_and_columns([0],[0]).det()
U_init.matrix_from_rows_and_columns([0,1],[0,1]).det()
U_init.matrix_from_rows_and_columns([0,1,2],[0,1,2]).det()
U_init.matrix_from_rows_and_columns([0,1,2,3],[0,1,2,3]).det()

y1*y3*y4*y5*y7*y8*y9*y10*y11*y13*y14*y15*y16*y17*y18 y1^2*y2*y3^2*y4^2*y5*y7*y8^2*y9^2*y10^2*y11*y13*y14^2*y15^2*y16^2*y17^2*y18 y1^3*y2^2*y3^3*y4^3*y5^2*y6*y7^2*y8^3*y9^3*y10^2*y11*y13*y14^2*y15^3*y16^3*y17^2*y18 y1^4*y2^3*y3^4*y4^4*y5^3*y6^2*y7^3*y8^4*y9^4*y10^3*y11^2*y12*y13^2*y14^3*y15^4*y16^3*y17^2*y18

U_init.matrix_from_rows_and_columns([1],[3]).det()

-y1*y2*y3*y7*y8*y13*y14*y15*y16*y17*y18 - y1*y2*y3*y7*y8*y13*y14*y15*y16*y17 - y1*y3*y7*y8*y13*y14*y15*y16*y17*y18 - y1*y2*y3*y7*y8*y14*y15*y16*y17 - y1*y3*y7*y8*y13*y14*y15*y16*y17 - y1*y2*y3*y7*y8*y14*y15*y16 - y1*y3*y7*y8*y14*y15*y16*y17 - y1*y3*y7*y8*y14*y15*y16 - y1*y3*y8*y14*y15*y16*y17 - y1*y3*y8*y14*y15*y16 - y1*y3*y8*y15*y16 - y1*y3*y8*y15

for i1 in range(4):
        for j1 in range(4):
                if how_long(upper_path.matrix_from_rows_and_columns([i1,],[j1]).det()) ==1:
                    print upper_path.matrix_from_rows_and_columns([i1,],[j1]).det()

1 y1*y2*y3 y1*y2*y3*y4*y5*y6*y7*y8 y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y12*y13*y14*y15

for i1 in range(4):
    for i2 in range(i1):
        for j1 in range(4):
            for j2 in range(j1):
                if how_long(upper_path.matrix_from_rows_and_columns([i1,i2],[j1,j2]).det()) ==1:
                    print upper_path.matrix_from_rows_and_columns([i1,i2],[j1,j2]).det()

y1*y2*y3 y1*y2*y3*y4*y5*y6*y7*y8 y1^2*y2^2*y3^2*y4*y5*y6*y7*y8 y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y12*y13*y14*y15 y1^2*y2^2*y3^2*y4*y5*y6*y7*y8*y9*y10*y11*y12*y13*y14*y15 y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9*y10*y11*y12*y13*y14*y15

upper_gsv = mutate_matrix(initial_feed,gsv_muts,upper_path)
lower_gsv = mutate_matrix(initial_feed,gsv_muts,lower_path)

upper_gsv[1,0]/upper_gsv[0,0]
upper_gsv[2,1]/upper_gsv[1,1]
upper_gsv[3,2]/upper_gsv[2,2]

y1 y2*y4 + y4 y5*y7*y9 + y5*y9 + y9

how_long(lower_gsv[3,2]/lower_gsv[2,2])

30

upper_gsv.matrix_from_rows_and_columns([0,1],[0,1]).det()

y1*y3

U_gsv = upper_gsv*w*lower_gsv*w.inverse()
#s_U_gsv = mutate_matrix(initial_feed,gsv_muts,s_U_init)

upper_gsv[0,0]
upper_gsv[1,1]
upper_gsv[2,2]
upper_gsv[3,3]

1 y1*y3 y1*y2*y3*y4*y8 y1*y2*y3*y4*y5*y7*y8*y9*y15

(w*lower_gsv*w.inverse())[0,0]
(w*lower_gsv*w.inverse())[1,1]
(w*lower_gsv*w.inverse())[2,2]
(w*lower_gsv*w.inverse())[3,3]

y1*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y4*y5*y6^2*y7*y8*y9*y10*y11*y14*y15*y16 y9*y10*y14*y15 1

y1*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y1*y3*y4*y5*y6^2*y7*y8*y9*y10*y11*y14*y15*y16 y1*y2*y3*y4*y8*y9*y10*y14*y15 y1*y2*y3*y4*y5*y7*y8*y9*y15

cas1=upper_path[1,1]*(lower_path)[3,3]/(lower_path)[2,2]

cas2=upper_path[2,2]/upper_path[1,1]*lower_path[2,2]/lower_path[1,1]

cas3=(upper_path[3,3]/upper_path[2,2])*(lower_path)[1,1]

def build_phi_1_fn(mat=U_gsv):
    phi_1_mat=matrix(FractionField(R),3,3)
    phi_1_mat[0,0]=mat[1,0]
    phi_1_mat[1,0]=mat[2,0]
    phi_1_mat[2,0]=mat[3,0]
    phi_1_mat[0,1]=((mat)^2)[1,0]
    phi_1_mat[1,1]=((mat)^2)[2,0]
    phi_1_mat[2,1]=((mat)^2)[3,0]
    phi_1_mat[0,2]=((mat)^3)[1,0]
    phi_1_mat[1,2]=((mat)^3)[2,0]
    phi_1_mat[2,2]=((mat)^3)[3,0]
    return phi_1_mat.det()

def build_phi_2_fn(mat=U_gsv):
    phi_1_mat=matrix(FractionField(R),3,3)
    phi_1_mat[0,0]=mat[1,0]
    phi_1_mat[1,0]=mat[2,0]
    phi_1_mat[2,0]=mat[3,0]
    phi_1_mat[0,1]=mat[1,1]
    phi_1_mat[1,1]=mat[2,1]
    phi_1_mat[2,1]=mat[3,1]
    phi_1_mat[0,2]=((mat)^2)[1,0]
    phi_1_mat[1,2]=((mat)^2)[2,0]
    phi_1_mat[2,2]=((mat)^2)[3,0]
    return phi_1_mat.det()

def build_phi_3_fn(mat=U_gsv):
    phi_1_mat=matrix(FractionField(R),2,2)
    phi_1_mat[0,0]=mat[2,0]
    phi_1_mat[1,0]=mat[3,0]
    phi_1_mat[0,1]=((mat)^2)[2,0]
    phi_1_mat[1,1]=((mat)^2)[3,0]
    return phi_1_mat.det()

d1=upper_gsv[0,0]*(w*lower_gsv*w.inverse())[0,0]
d2=upper_gsv[1,1]*(w*lower_gsv*w.inverse())[1,1]
d3=upper_gsv[2,2]*(w*lower_gsv*w.inverse())[2,2]
d4=upper_gsv[3,3]*(w*lower_gsv*w.inverse())[3,3]


h44=U_gsv.matrix_from_rows_and_columns([0],[0]).det()
h34=U_gsv.matrix_from_rows_and_columns([1],[0]).det()
h24=U_gsv.matrix_from_rows_and_columns([2],[0]).det()
h14=U_gsv.matrix_from_rows_and_columns([3],[0]).det()
h33=U_gsv.matrix_from_rows_and_columns([0,1],[0,1]).det()
h23=U_gsv.matrix_from_rows_and_columns([1,2],[0,1]).det()
h13=U_gsv.matrix_from_rows_and_columns([2,3],[0,1]).det()

h22=U_gsv.matrix_from_rows_and_columns([0,1,2],[0,1,2]).det()
h12=U_gsv.matrix_from_rows_and_columns([1,2,3],[0,1,2]).det()

f21 =build_phi_3_fn()

f11 = build_phi_1_fn()

f12 = build_phi_2_fn()

h44
h34
h33
h24
h23
h22
h14
h13
h12

y1*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y1^2*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y1^2*y2^2*y3^2*y4^2*y5^3*y6^4*y7^3*y8^2*y9^2*y10^3*y11^4*y12*y13*y14^3*y15^2*y16^4*y17*y18 y1^2*y2^2*y3*y4^2*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y1^3*y2^3*y3^2*y4^3*y5^3*y6^4*y7^3*y8^2*y9^2*y10^3*y11^4*y12*y13*y14^3*y15^2*y16^4*y17*y18 y1^3*y2^3*y3^3*y4^3*y5^3*y6^4*y7^3*y8^3*y9^3*y10^4*y11^4*y12*y13*y14^4*y15^3*y16^4*y17*y18 y1^2*y2^2*y3*y4^2*y5^2*y6^2*y7^2*y8*y9^2*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y1^3*y2^3*y3^2*y4^4*y5^4*y6^4*y7^3*y8^2*y9^3*y10^3*y11^4*y12*y13*y14^3*y15^2*y16^4*y17*y18 y1^4*y2^4*y3^3*y4^4*y5^4*y6^4*y7^4*y8^3*y9^4*y10^4*y11^4*y12*y13*y14^4*y15^3*y16^4*y17*y18

f21
f12
f11

y1^5*y2^5*y3^3*y4^5*y5^5*y6^5*y7^5*y8^3*y9^4*y10^4*y11^6*y12^2*y13^2*y14^5*y15^3*y16^6*y17^2*y18^2 -y1^6*y2^6*y3^4*y4^6*y5^6*y6^6*y7^6*y8^4*y9^5*y10^5*y11^7*y12^2*y13^2*y14^6*y15^4*y16^7*y17^2*y18^2 y1^9*y2^9*y3^6*y4^8*y5^8*y6^8*y7^9*y8^6*y9^7*y10^7*y11^9*y12^3*y13^3*y14^9*y15^6*y16^10*y17^3*y18^3

d1
d2
d3
d4

y1*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12*y13*y14^2*y15*y16^3*y17*y18 y1*y3*y4*y5*y6^2*y7*y8*y9*y10*y11*y14*y15*y16 y1*y2*y3*y4*y8*y9*y10*y14*y15 y1*y2*y3*y4*y5*y7*y8*y9*y15

d2== h33/h44
d3 == h22/h33
d1 == h44

True True True

#### so d4

def build_pb_mat(fn_list,mat):
    res=matrix(QQ,len(fn_list))
    for i,f in enumerate(fn_list):
        for j,g in enumerate(fn_list):
            res[i,j]=PB(f,g,mat=mat)/(f*g)
    return res

build_pb_mat(cluster_fn,gsv_mat)

[ 0 1/2 1/2 1 0 1/2 1 3/2 0 1 3/2 2] [-1/2 0 1/2 1 1 1 2 2 3/2 2 5/2 3] [-1/2 -1/2 0 1 0 1/2 1 3/2 3/2 2 2 3] [ -1 -1 -1 0 -1 -1 0 0 -1 0 0 0] [ 0 -1 0 1 0 0 2 1 0 1 1 0] [-1/2 -1 -1/2 1 0 0 2 1 3/2 2 3/2 1] [ -1 -2 -1 0 -2 -2 0 -1 -1 0 -1 -2] [-3/2 -2 -3/2 0 -1 -1 1 0 1/2 2 1/2 1] [ 0 -3/2 -3/2 1 0 -3/2 1 -1/2 0 1 -1/2 -2] [ -1 -2 -2 0 -1 -2 0 -2 -1 0 -2 -4] [-3/2 -5/2 -2 0 -1 -3/2 1 -1/2 1/2 2 0 -1] [ -2 -3 -3 0 0 -1 2 -1 2 4 1 0]

cluster_fn = [h44,h34,h24,h14,h33,h23,h13,f21,h22,h12,f12,f11]

exchange_quiver=matrix([[0,-1,0,0,0,0,0,0,0,0,0,0],[1,0,-1,0,-1,1,0,0,0,0,0,0],[0,1,0,-1,0,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,1,-1,0],[0,1,0,0,0,-1,0,0,0,0,0,0],[0,-1,1,0,1,0,-1,0,-1,1,0,0],[0,0,-1,0,0,1,0,1,0,0,-1,0],[0,0,0,0,0,0,-1,0,0,0,2,-1],[0,0,0,0,0,1,0,0,0,-1,0,0],[0,0,0,-1,0,-1,0,0,1,0,1,0],[0,0,0,1,0,0,1,-2,0,-1,0,1],[0,0,0,0,0,0,0,1,0,0,-1,0]]);

mod_exchange_quiver = matrix(QQ,[[0,-1,0,0,0,0,0,0,0,0,0,0],[1,0,-1,0,-1,1,0,0,0,0,0,0],[0,1,0,-1,0,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,1,-1,0],[0,1,0,0,0,-1,0,0,0,0,0,0],[0,-1,1,0,1,0,-1,0,-1,1,0,0],[0,0,-1,0,0,1,0,1,0,0,-1,0],[0,0,0,0,0,0,-1,0,0,0,2,-1],[0,0,0,0,0,1,0,0,0,-1,0,0],[0,0,0,-1,0,-1,0,0,1,0,1,0],[0,0,0,1,0,0,1,-2,0,-1,0,1],[0,0,0,0,0,0,0,1,0,0,-1,0]]);

mod_exchange_quiver[0,4]=1/2
mod_exchange_quiver[4,0]=-1/2
mod_exchange_quiver[4,8]=1/2
mod_exchange_quiver[8,4]=-1/2

U_gsv.characteristic_polynomial()

x^4 + (y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y12*y14*y15*y16*y18 + y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y12*y14*y15*y16 + y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y13*y14*y15*y16 + y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y14*y15*y16*y17)*x^3 + (y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9^2*y10^2*y11^2*y12^2*y14^2*y15^2*y16^2*y18 + y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9^2*y10^2*y11^2*y12*y13*y14^2*y15^2*y16^2*y18 + y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9^2*y10^2*y11^2*y12*y14^2*y15^2*y16^2*y17*y18 + y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9^2*y10^2*y11^2*y12*y13*y14^2*y15^2*y16^2 + y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9^2*y10^2*y11^2*y12*y14^2*y15^2*y16^2*y17 + y1^2*y2^2*y3^2*y4^2*y5^2*y6^2*y7^2*y8^2*y9^2*y10^2*y11^2*y13*y14^2*y15^2*y16^2*y17)*x^2 + (y1^3*y2^3*y3^3*y4^3*y5^3*y6^3*y7^3*y8^3*y9^3*y10^3*y11^3*y12^2*y13*y14^3*y15^3*y16^3*y18 + y1^3*y2^3*y3^3*y4^3*y5^3*y6^3*y7^3*y8^3*y9^3*y10^3*y11^3*y12^2*y14^3*y15^3*y16^3*y17*y18 + y1^3*y2^3*y3^3*y4^3*y5^3*y6^3*y7^3*y8^3*y9^3*y10^3*y11^3*y12*y13*y14^3*y15^3*y16^3*y17*y18 + y1^3*y2^3*y3^3*y4^3*y5^3*y6^3*y7^3*y8^3*y9^3*y10^3*y11^3*y12*y13*y14^3*y15^3*y16^3*y17)*x + y1^4*y2^4*y3^4*y4^4*y5^4*y6^4*y7^4*y8^4*y9^4*y10^4*y11^4*y12^2*y13*y14^4*y15^4*y16^4*y17*y18

#11: C11*(l1-l2)*...(l1-l4)
#12:C41

### Vertices on left: all same thing, det C
### U = C^-1 * L * C

### Vertices of splitback quiver:
### 16: C[2,3,4],[1,2,3]
### 17: C[3,4],[1,2]
### 18: C[4],[1]
### 12: C[4],[4]
### 6:  C[3,4][3,4]
### 2:  C[2,3,4],[2,3,4]
### 10: C[2,3,4],[1,2,4]
### 5:  C[2,3,4],[1,3,4]
### 13: C[3,4],[L*C_col1, C_col4]
### 14: L*C_col1, L*C_col2,C_col4, rows=[2,3,4]
### 7:  L*C_col1, L*C_col3, L*C_col4, rows=[2,3,4]
### 3: [2,3,4,L*C_col1]
### 8: [3,4, L*Col_2, L*col1]
### 15: [4, L*Co1, L*Col2, L*Col3]

exchange_quiver.kernel()

Free module of degree 12 and rank 0 over Integer Ring Echelon basis matrix: []

def build_X_coords(mat=exchange_quiver,fns=cluster_fn):
    res=[]
    for i in range(len(cluster_fn)):
        toAdd=1
        for j,f in enumerate(cluster_fn):
            toAdd=toAdd*f^(mat[i,j])
        res.append(toAdd)
    return res

exchange_quiver

[ 0 -1 0 0 0 0 0 0 0 0 0 0] [ 1 0 -1 0 -1 1 0 0 0 0 0 0] [ 0 1 0 -1 0 -1 1 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 1 -1 0] [ 0 1 0 0 0 -1 0 0 0 0 0 0] [ 0 -1 1 0 1 0 -1 0 -1 1 0 0] [ 0 0 -1 0 0 1 0 1 0 0 -1 0] [ 0 0 0 0 0 0 -1 0 0 0 2 -1] [ 0 0 0 0 0 1 0 0 0 -1 0 0] [ 0 0 0 -1 0 -1 0 0 1 0 1 0] [ 0 0 0 1 0 0 1 -2 0 -1 0 1] [ 0 0 0 0 0 0 0 1 0 0 -1 0]

Xs=build_X_coords()
modXs=build_X_coords(mat=-2*mod_exchange_quiver)

for v in Xs:
    print v

1/(y1^2*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12^2*y13*y14^2*y15*y16^3*y17*y18) y2 y5 -y10 1/(y1*y2*y3*y4^2*y5*y6^2*y7*y8*y9*y10*y11*y14*y15*y16) y7 -y6 y11 1/(y1*y2*y3*y4*y5*y7*y8*y9^2*y10*y14*y15) -y1^4*y2^4*y3^4*y4^4*y5^4*y6^4*y7^4*y8^4*y9^4*y10^4*y11^4*y12^2*y13*y14^5*y15^4*y16^4*y17*y18 y16 1/(-y1*y2*y3*y4*y5*y6*y7*y8*y9*y10*y11*y14*y15*y16)

h33

y1^2*y2^2*y3^2*y4^2*y5^3*y6^4*y7^3*y8^2*y9^2*y10^3*y11^4*y12^2*y13*y14^3*y15^2*y16^4*y17*y18

### some observations

d2/(Xs[0]*h33)

y1

1/(Xs[4]*d2)

y2*y4

1/(Xs[8]*d3)

y5*y7*y9

Xs[9]/(d3^2*h33)

(-y5*y7)/y10

Xs[11]*d2

y6/(-y2)

X_pb_mat =build_pb_mat(Xs)
mod_X_pb_mat = build_pb_mat(modXs)

X_pb_mat.matrix_from_rows_and_columns([1,2,5,3,6,9,10,7,11],[1,2,5,3,6,9,10,7,11])+exchange_quiver.matrix_from_rows_and_columns([1,2,5,3,6,9,10,7,11],[1,2,5,3,6,9,10,7,11])

[0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0]

mod_X_pb_mat/4

[ 0 1 0 0 -1/2 0 0 0 1 -2 0 1/2] [ -1 0 1 0 1 -1 0 0 0 0 0 0] [ 0 -1 0 1 0 1 -1 0 0 0 0 0] [ 0 0 -1 0 0 0 0 0 0 -1 1 0] [ 1/2 -1 0 0 0 1 0 0 -1/2 0 0 0] [ 0 1 -1 0 -1 0 1 0 1 -1 0 0] [ 0 0 1 0 0 -1 0 -1 0 0 1 0] [ 0 0 0 0 0 0 1 0 0 0 -2 1] [ -1 0 0 0 1/2 -1 0 0 0 3 0 -1/2] [ 2 0 0 1 0 1 0 0 -3 0 -1 0] [ 0 0 0 -1 0 0 -1 2 0 1 0 -1] [-1/2 0 0 0 0 0 0 -1 1/2 0 1 0]

X_pb_mat + exchange_quiver

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'X_pb_mat' is not defined

Kcas1=upper_gsv[1,1]*(lower_gsv)[3,3]/(lower_gsv)[2,2]
Kcas2=(upper_gsv[2,2]/upper_gsv[1,1])*(lower_gsv[2,2]/lower_gsv[1,1])

Kcas3=(upper_gsv[3,3]/upper_gsv[2,2])*(lower_gsv)[1,1]

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'upper_gsv' is not defined

Kcas1
Kcas2
Kcas3

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'Kcas1' is not defined

f2

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'f2' is not defined

U_gsv.det()

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'U_gsv' is not defined

f1/(Kcas1*Kcas2*Kcas3)
f2/(Kcas1*Kcas2*Kcas3)
f3/(Kcas1*Kcas2*Kcas3)
f4/(Kcas1*Kcas2*Kcas3)

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'f1' is not defined

cf(f1,f5)

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'cf' is not defined

z=to_mon(v)^2

#s_U_gsv.matrix_from_rows_and_columns([0,1],[0,1]).det()
#s_U_gsv.matrix_from_rows_and_columns([1,2],[0,1]).det()
#s_U_gsv.matrix_from_rows_and_columns([2,3],[0,1]).det()

#s_U_gsv.matrix_from_rows_and_columns([0,1,2],[0,1,2]).det()
#s_U_gsv.matrix_from_rows_and_columns([1,2,3],[0,1,2]).det()

def build_phi_1_fn(mat=U_gsv):
    phi_1_mat=matrix(FractionField(R),3,3)
    phi_1_mat[0,0]=mat[1,0]
    phi_1_mat[1,0]=mat[2,0]
    phi_1_mat[2,0]=mat[3,0]
    phi_1_mat[0,1]=((mat)^2)[1,0]
    phi_1_mat[1,1]=((mat)^2)[2,0]
    phi_1_mat[2,1]=((mat)^2)[3,0]
    phi_1_mat[0,2]=((mat)^3)[1,0]
    phi_1_mat[1,2]=((mat)^3)[2,0]
    phi_1_mat[2,2]=((mat)^3)[3,0]
    return phi_1_mat.det()

def build_phi_2_fn(mat=U_gsv):
    phi_1_mat=matrix(FractionField(R),3,3)
    phi_1_mat[0,0]=mat[1,0]
    phi_1_mat[1,0]=mat[2,0]
    phi_1_mat[2,0]=mat[3,0]
    phi_1_mat[0,1]=mat[1,1]
    phi_1_mat[1,1]=mat[2,1]
    phi_1_mat[2,1]=mat[3,1]
    phi_1_mat[0,2]=((mat)^2)[1,0]
    phi_1_mat[1,2]=((mat)^2)[2,0]
    phi_1_mat[2,2]=((mat)^2)[3,0]
    return phi_1_mat.det()

def build_phi_3_fn(mat=U_gsv):
    phi_1_mat=matrix(FractionField(R),2,2)
    phi_1_mat[0,0]=mat[2,0]
    phi_1_mat[1,0]=mat[3,0]
    phi_1_mat[0,1]=((mat)^2)[2,0]
    phi_1_mat[1,1]=((mat)^2)[3,0]
    return phi_1_mat.det()

phi1= build_phi_1_fn()
phi2=build_phi_2_fn()
phi3=build_phi_3_fn()

cluster_fn=[f1,f2,f3,f4,f5,phi1,phi2,phi3]

def build_pb_mat(fn_list):
    res=matrix(QQ,len(fn_list))
    for i,f in enumerate(fn_list):
        for j,g in enumerate(fn_list):
            res[i,j]=cf(f,g)
    return res

build_pb_mat(cluster_fn)

[ 0 0 2 0 1 0 1 1] [ 0 0 2 3/2 2 1 3/2 1] [ -2 -2 0 -1 0 -2 -1 -1] [ 0 -3/2 1 0 1 -2 -1/2 -1/2] [ -1 -2 0 -1 0 -4 -2 -2] [ 0 -1 2 2 4 0 1 -1] [ -1 -3/2 1 1/2 2 -1 0 -1/2] [ -1 -1 1 1/2 2 1 1/2 0]

w

[ 0 0 0 -1] [ 0 0 1 0] [ 0 -1 0 0] [ 1 0 0 0]

lower_gsv[,1]/lower_gsv[1,1]

y5*y6^2*y7*y8*y11*y16 + y6^2*y7*y8*y11*y16 + y6*y7*y8*y11*y16 + y6*y7*y8*y11 + y6*y7*y8 + y7*y8 + y8

U_gsv = upper_gsv*w*lower_gsv*(w.inverse())

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'upper_gsv' is not defined

U_gsv[0,0]
U_gsv[1,0]
U_gsv[2,0]
U_gsv[3,0]

y1*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12^2*y13*y14^2*y15*y16^3*y17*y18 y1^2*y2^2*y3*y4*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12^2*y13*y14^2*y15*y16^3*y17*y18 y1^2*y2^2*y3*y4^2*y5^2*y6^2*y7^2*y8*y9*y10^2*y11^3*y12^2*y13*y14^2*y15*y16^3*y17*y18 y1^2*y2^2*y3*y4^2*y5^2*y6^2*y7^2*y8*y9^2*y10^2*y11^3*y12^2*y13*y14^2*y15*y16^3*y17*y18

upper_gsv.matrix_from_rows_and_columns([2,3],[0,1]).det()

y1^2*y2*y3*y4^2*y5*y9

get_path(lower_path[3,2]/lower_path[2,2])

[y3, y7, y13, y18, y11, y5]

pretty_print(upper_path.subs({y2:-1,y6:-1,y12:-1,y5:1/y7,y3:-1/y1,y4:-1/y8,y11:1/y13,y10:1/y14,y9:-1/y15}))

$\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

upper_path.subs({y2:-1,y6:-1})[3,0].factor()

(y12 + 1) * y11 * y10 * y9 * y5 * y4 * y1

### how to get E1:
E1 = upper_path[1,0]/upper_path[0,0]

### how to get E2:
E2 = upper_path[2,1]/upper_path[1,1]

### how to get E3:
E3 = upper_path[3,2]/upper_path[2,2]

E3

y9*y10*y11*y12*y13*y14 + y9*y10*y11*y12*y13 + y9*y10*y11*y12 + y9*y10*y11 + y9*y10 + y9

get_path(E1)
get_path(E2)
get_path(E3)

[y1, y2] [y4, y5, y6, y7] [y9, y10, y11, y12, y13, y14]

### Now let's self-fold:

## The path for E's:
short_path = rect(y1,y4,y9)*hexa(y5,y2,y7)*rect(y3,y8,y15)

## folded E1
e1 = short_path[1,0]/short_path[0,0]

## folded E2
e2 = short_path[2,1]/short_path[1,1]

## folded E3
e3 = short_path[3,2]/short_path[2,2]

e1,e2,e3

(y1, y2*y4 + y4, y5*y7*y9 + y5*y9 + y9)

### the path for F's

long_path = rect(y15,y8,y3)*hexa(y2,y7,y5)*rect(y14,y6,y10)*hexa(y11,y13,y12)*rect(y16,y17,y18)*hexa(y12,y11,y13)*rect(y10,y6,y14)*hexa(y7,y5,y2)*rect(y9,y4,y1)

## folded F3
f3 = long_path[1,0]/long_path[0,0]

## folded F2
f2 = long_path[2,1]/long_path[1,1]

## folded F3
f1 = long_path[3,2]/long_path[2,2]

how_long(f1),how_long(f2)

(12, 8)

F1

[1 0 0 0] [1 1 0 0] [0 0 1 0] [0 0 0 1]

geom_folding_muts=[2, 6, (12,18), 5, 7, 11, 13, 10, 6, 14]
geom_gsv = [(17,12),16,11,(13,17)]

folded_feed=build_feed(initial_feed,geom_folding_muts)
step1_feed = build_feed(folded_feed,[(17,12)])

how_long(folded_feed.inv_mutation_auto([(17,12)],f1))

16

gsv_f1 = folded_feed.inv_mutation_auto(geom_gsv,f1)
gsv_f2 = folded_feed.inv_mutation_auto(geom_gsv,f2)
gsv_f3 = folded_feed.inv_mutation_auto(geom_gsv,f3)

how_long(folded_feed.inv_mutation_auto(geom_gsv[:],f2))

7

cap=rect(y14,y6,y10)*hexa(y11,y13,y12)*rect(y16,y17,y18)*hexa(y12,y11,y13)*rect(y10,y6,y14)

hol = rect(y16,y17,y18)*hexa(y12,y11,y13)

h1 = hol.trace()

h2 = ((hol.trace())^2 - (hol^2).trace())/2

y11*y12*y13*y16*y17*y18 + y11*y16*y17 + y16 + 1

(y11^3*y12*y13^6*y16^2*y17^2*y18 + 2*y11^3*y12*y13^5*y16^2*y17^2*y18 + y11^3*y12*y13^5*y16*y17^2*y18 + y11^3*y12*y13^4*y16^2*y17^2*y18 + y11^3*y13^6*y16^2*y17 + y11^3*y12*y13^4*y16*y17^2*y18 + y11^2*y12*y13^5*y16*y17^2*y18 + 2*y11^3*y13^5*y16^2*y17 + 2*y11^2*y12*y13^4*y16*y17^2*y18 + 2*y11^3*y13^5*y16*y17 + y11^3*y13^4*y16^2*y17 + y11^2*y12*y13^3*y16*y17^2*y18 + 2*y11^3*y13^4*y16*y17 + 3*y11^2*y13^5*y16*y17 + y11^3*y13^4*y17 + 6*y11^2*y13^4*y16*y17 + y11*y13^5*y16*y17 + 2*y11^2*y13^4*y17 + 3*y11^2*y13^3*y16*y17 + 3*y11*y13^4*y16*y17 + y11*y13^4*y16 + 2*y11^2*y13^3*y17 + y11*y13^4*y17 + 3*y11*y13^3*y16*y17 + 3*y11*y13^3*y16 + 2*y11*y13^3*y17 + y11*y13^2*y16*y17 + 3*y11*y13^2*y16 + y11*y13^2*y17 + y11*y13*y16)/(y11^3*y12*y13^6*y16^2*y17^2*y18 + 2*y11^3*y12*y13^5*y16^2*y17^2*y18 + 2*y11^3*y12*y13^5*y16*y17^2*y18 + y11^3*y12*y13^4*y16^2*y17^2*y18 + 2*y11^3*y12*y13^4*y16*y17^2*y18 + 3*y11^2*y12*y13^5*y16*y17^2*y18 + y11^3*y12*y13^4*y17^2*y18 + 6*y11^2*y12*y13^4*y16*y17^2*y18 + y11*y12*y13^5*y16*y17^2*y18 + y11^3*y13^5*y16*y17 + 2*y11^2*y12*y13^4*y17^2*y18 + 3*y11^2*y12*y13^3*y16*y17^2*y18 + 3*y11*y12*y13^4*y16*y17^2*y18 + y11^3*y13^4*y16*y17 + 2*y11^2*y13^5*y16*y17 + y11*y12*y13^4*y16*y17*y18 + 2*y11^2*y12*y13^3*y17^2*y18 + y11*y12*y13^4*y17^2*y18 + 3*y11*y12*y13^3*y16*y17^2*y18 + y11^3*y13^4*y17 + 4*y11^2*y13^4*y16*y17 + y11*y13^5*y16*y17 + 3*y11*y12*y13^3*y16*y17*y18 + 2*y11*y12*y13^3*y17^2*y18 + y11*y12*y13^2*y16*y17^2*y18 + 3*y11^2*y13^4*y17 + 2*y11^2*y13^3*y16*y17 + 3*y11*y13^4*y16*y17 + 3*y11*y12*y13^2*y16*y17*y18 + y11*y12*y13^2*y17^2*y18 + y11*y13^4*y16 + 3*y11^2*y13^3*y17 + 3*y11*y13^4*y17 + 3*y11*y13^3*y16*y17 + y11*y12*y13*y16*y17*y18 + 3*y11*y13^3*y16 + 6*y11*y13^3*y17 + y13^4*y17 + y11*y13^2*y16*y17 + y11*y13^3 + 3*y11*y13^2*y16 + 3*y11*y13^2*y17 + 3*y13^3*y17 + 2*y11*y13^2 + y13^3 + y11*y13*y16 + 3*y13^2*y17 + y11*y13 + 3*y13^2 + y13*y17 + 3*y13 + 1)

hol.det()

y11^2*y12*y13*y16^3*y17^2*y18

get_path(cap[1,0])
get_path(cap[2,1]/cap[1,1])
get_path(cap[3,2]/cap[2,2])

[y14, y16] [y6, y13, y17, y11] [y10, y11, y12, y18, y12, y13]

cap[3,2].factor()

y17 * y16 * y14 * y13 * y11 * y10^2 * y6^2 * (y11*y12^2*y13*y18 + y11*y12^2*y18 + y11*y12*y18 + y11*y12 + y11 + 1)

10 12 17

pre = y10^2*y6^2*y11*y16*y14

from_network = pre * (1 + y16 + y16*y13 + y16*y13*y17*(1+y13) +y16*y13*y17*y12*(1+y13) 
                     + y16*y13^2*y17^2*y12*(1+y13+y13*y11) + y16*y13^2*y17^2*y12^2*y18*(1+y13+y13*y11) 
                     + y16*y13^2*y17^2*y12*y18*(1+y13+y13*y11) + y16*y13*y17*y12*y18*(1+y13) 
                     + y16*y13^3*y17^3*y12^2*y18*(1+y11) + y16*y13^4*y17^3*y12^2*y18*(1+y11)^2
                     + y16^2*y13^4*y17^3*y12^2*y18*y11^2)

how_long(from_network)

24

from_network == gsv_cap[3,2]

True

show_monomials(from_network)

y6^2*y10^2*y11^3*y12^2*y13^4*y14*y16^3*y17^3*y18 y6^2*y10^2*y11^3*y12^2*y13^4*y14*y16^2*y17^3*y18 2*y6^2*y10^2*y11^2*y12^2*y13^4*y14*y16^2*y17^3*y18 y6^2*y10^2*y11^2*y12^2*y13^3*y14*y16^2*y17^3*y18 y6^2*y10^2*y11*y12^2*y13^4*y14*y16^2*y17^3*y18 y6^2*y10^2*y11^2*y12^2*y13^3*y14*y16^2*y17^2*y18 y6^2*y10^2*y11*y12^2*y13^3*y14*y16^2*y17^3*y18 y6^2*y10^2*y11^2*y12*y13^3*y14*y16^2*y17^2*y18 y6^2*y10^2*y11*y12^2*y13^3*y14*y16^2*y17^2*y18 y6^2*y10^2*y11^2*y12*y13^3*y14*y16^2*y17^2 y6^2*y10^2*y11*y12^2*y13^2*y14*y16^2*y17^2*y18 y6^2*y10^2*y11*y12*y13^3*y14*y16^2*y17^2*y18 y6^2*y10^2*y11*y12*y13^3*y14*y16^2*y17^2 y6^2*y10^2*y11*y12*y13^2*y14*y16^2*y17^2*y18 y6^2*y10^2*y11*y12*y13^2*y14*y16^2*y17^2 y6^2*y10^2*y11*y12*y13^2*y14*y16^2*y17*y18 y6^2*y10^2*y11*y12*y13^2*y14*y16^2*y17 y6^2*y10^2*y11*y12*y13*y14*y16^2*y17*y18 y6^2*y10^2*y11*y12*y13*y14*y16^2*y17 y6^2*y10^2*y11*y13^2*y14*y16^2*y17 y6^2*y10^2*y11*y13*y14*y16^2*y17 y6^2*y10^2*y11*y13*y14*y16^2 y6^2*y10^2*y11*y14*y16^2 y6^2*y10^2*y11*y14*y16

how_long(gsv_cap[3,0])
how_long(gsv_cap[3,1])
how_long(gsv_cap[3,2])
how_long(gsv_cap[3,3])
         

7 24 24 1

how_long(gsv_cap[3,2]/gsv_cap[2,2])

24

cap[1,0]
(cap[2,1]/cap[1,1])
(cap[3,2]/cap[2,2])

y14*y16 + y14 y6*y11*y13*y17 + y6*y13*y17 + y6*y13 + y6 y10*y11*y12^2*y13*y18 + y10*y11*y12^2*y18 + y10*y11*y12*y18 + y10*y11*y12 + y10*y11 + y10

folded_feed.inv_mutation_auto([(17,12)],cap[1,0])
folded_feed.inv_mutation_auto([(17,12)],cap[2,1]/cap[1,1])
folded_feed.inv_mutation_auto([(17,12)],cap[3,2]/cap[2,2])

y14*y16 + y14 y6*y11*y13 + y6*y13 + y6 y10*y11*y12^2*y13*y17^3*y18 + y10*y11*y12^2*y17^3*y18 + y10*y11*y12^2*y17^2*y18 + y10*y11*y12*y17^2*y18 + y10*y11*y12*y17^2 + y10*y11*y12*y17*y18 + y10*y11*y12*y17 + y10*y11*y17 + y10*y11 + y10

### Now we have matrix formulas for the self-folded F's we can translate them into dimer counts etc.  

folded_mat = H3(y3)*H2(y8)*H1(y15)*F3*H3(y2)*F2*H2(y7)*F3*H3(y5)*F1*F2*H2(y6)*F3*H3(y10)* F3*H3(y11)*F2*H2(y13)*F3*H3(y12)*H1(y14)*F1*F2*H2(y17)*F3 *H3(y18)* F3*H3(y12)*F2*H2(y11)*F3*H3(y13)*H1(y16)*F1*F2*H2(y6)*F3 *H3(y14)*F3*H3(y7)*F2*H2(y5)*F3*H3(y2)*H1(y10)*F1*F2*F3*H1(y9)*H2(y4)*H3(y1)

step1 =  H3(y3)*H2(y8)*H1(y15)*F3*H3(y2)*F2*H2(y7)*F3*H3(y5)*F1*H1(y14)*F2*H2(y6)*F3*H3(y10)* F3*H3(y11)*F2*H2(y13)*F3*H3(y17)*F3*H3(y12)*F1*H1(y16)*F3 *H3(y18)*F3*H3(y17/y18)*F2*H2(y11)* F3*H3(y18)*F3*H3(y12)*F3*H3(y17)*F3*H3(y13)*F1*H1(y10)*F2*H2(y6)*F3 *H3(y14)*F3*H3(y7)*F2*H2(y5)*F3*H3(y2)*F1*F2*F3*H1(y9)*H2(y4)*H3(y1)

folded_mat==long_path

True

step1 == mutate_matrix(folded_feed,[(17,12)],folded_mat)

True

step2 = H3(y3)*H2(y8)*H1(y15)*F3*H3(y2)*F2*H2(y7)*F3*H3(y5)*F1*H1(y14)*F2*H2(y6)*F3*H3(y10)*F3*H3(y16)*F3*H3(y11)* F2*H2(y13)*F3*H3(y17)*F3*H3(y12)*F3 *H3(y18)*F3*H3(y17/y18)*F2*H2(y16)*F1*H1(y10)*F2*H2(y11)* F3*H3(y18)*F3*H3(y12)*F3*H3(y17)*F3*H3(y13)*F2*H2(y6)*F3*H3(y16)*F3 *H3(y14)*F3*H3(y7)*F2*H2(y5)*F3*H3(y2)*F1*F2*F3*H1(y9)*H2(y4)*H3(y1)

mutated2 = mutate_matrix(folded_feed,[(17,12),16],folded_mat)
mutated3 = mutate_matrix(folded_feed,[(17,12),16,11],folded_mat)

step2==mutated2

True

get_path(mutated3[1,0]/mutated3[0,0])

[y15, y14, y10]

get_path(mutated3[2,1]/mutated3[1,1])

[y8, y7, y6, y11, y13, y16, y6, y5]

show_monomials((mutated3[3,2]/mutated3[2,2]).subs({y11:0}))

y2*y3*y5*y10*y12^2*y16*y17^3*y18 y2*y3*y5*y10*y12^2*y16*y17^2*y18 y2*y3*y5*y10*y12*y16*y17^2*y18 y2*y3*y5*y10*y12*y16*y17^2 y2*y3*y5*y10*y12*y16*y17*y18 y2*y3*y5*y10*y12*y16*y17 y2*y3*y5*y10*y16*y17 y2*y3*y5*y10*y16 y2*y3*y5*y10 y2*y3*y5 y2*y3 y3

mess=(mutated3[3,2]/mutated3[2,2])

show_monomials(diff(((1/2)*y12^2*diff(mess,y12,2)),y13))

y2^2*y3*y5*y7*y10*y11^2*y12^2*y14*y16^2*y17^3*y18 y2*y3*y5*y7*y10*y11^2*y12^2*y14*y16^2*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y14*y16^2*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y16^2*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y16*y17^3*y18 y2*y3*y5*y10*y11*y12^2*y16*y17^3*y18

show_monomials(((1/2)*y12^2*diff(mess,y12,2)/(y2*y3*y5*y10)).subs({y13:0}))

y11^2*y12^2*y16*y17^3*y18 2*y11*y12^2*y16*y17^3*y18 y11*y12^2*y16*y17^2*y18 y12^2*y16*y17^3*y18 y12^2*y16*y17^2*y18

show_monomials((mutated3[3,2]/mutated3[2,2]))

y2^2*y3*y5*y7*y10*y11^2*y12^2*y13*y14*y16^2*y17^3*y18 y2*y3*y5*y7*y10*y11^2*y12^2*y13*y14*y16^2*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y13*y14*y16^2*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y13*y16^2*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y13*y16*y17^3*y18 y2*y3*y5*y10*y11^2*y12^2*y16*y17^3*y18 y2*y3*y5*y10*y11*y12^2*y13*y16*y17^3*y18 2*y2*y3*y5*y10*y11*y12^2*y16*y17^3*y18 y2*y3*y5*y10*y11*y12^2*y16*y17^2*y18 y2*y3*y5*y10*y12^2*y16*y17^3*y18 y2*y3*y5*y10*y11*y12*y16*y17^2*y18 y2*y3*y5*y10*y12^2*y16*y17^2*y18 y2*y3*y5*y10*y11*y12*y16*y17^2 y2*y3*y5*y10*y12*y16*y17^2*y18 y2*y3*y5*y10*y12*y16*y17^2 y2*y3*y5*y10*y12*y16*y17*y18 y2*y3*y5*y10*y12*y16*y17 y2*y3*y5*y10*y16*y17 y2*y3*y5*y10*y16 y2*y3*y5*y10 y2*y3*y5 y2*y3 y3

get_path(mutated3[2,2])

[y4*y5*y6^2*y7*y8*y9*y10*y11*y13*y14*y15*y16]

gsv_mat =  mutate_matrix(folded_feed,[(17,12),16,11,13],folded_mat)

show_monomials(gsv_mat[3,2]/gsv_mat[2,2]

gg = H1(y1)*H2(y2)*H3(y3)*F1*H1(y4)*F2*H2(y5)*F3*H3(y6)*F1*H1(y7)*F2*H2(y8)*F1*H1(y9)

gg[0,0].factor()

1

##### folded quivermat (no dashed, don't care)
               
fC=matrix([[0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,0,-1,-1,1,0,-1,1,0,0,0,0,0,0,0,0,0,0],[-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,-1,0,1,0,-1,1,0,-1,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,-1,0,0,0,-1,0,1,0,0,0,0,0],[0,1,0,0,-1,1,0,-1,0,0,0,0,0,-1,1,0,0,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0],[0,0,0,0,-1,0,0,0,1,0,1,0,0,0,0,-1,0,0],[0,0,0,0,0,1,0,0,0,-1,0,1,-1,0,0,1,-1,0],[0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0],[0,0,0,0,0,-1,0,0,0,0,1,-1,0,1,0,-1,1,0],[0,0,0,0,0,0,1,0,0,0,0,0,-1,0,-1,1,0,0],[0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,-1,0,1,-1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]);