# SCRIPT: BH-KNOT-08
# Quantum invariants at roots of unity and the mirror. q = e^{2 pi i/N}.
#  Figure-eight 4_1: Kashaev invariant <4_1>_N = sum_{k=0}^{N-1} |(q;q)_k|^2 (real and positive).
#  Trefoil 3_1: the Kontsevich-Zagier series F(q) = sum_{n>=0} (q;q)_n, which terminates at a root of
#  unity and equals the Kashaev invariant of the trefoil up to conjugation.
# Kill conditions, written before the run:
#  K1  volume conjecture for 4_1: (2 pi/N) log <4_1>_N -> Vol(4_1) = 2.0298832128; with the known
#      subleading terms, (2 pi/N)(log<4_1>_N - 1.5 log N + 0.25 log 3) must be within 1e-2 of Vol at N = 800
#      and approach it monotonically. Kill otherwise.
#  K2  trefoil: |F(q)|/N^{3/2} converges (changes by less than 1e-3 from N = 400 to N = 800), with no
#      exponential growth (the trefoil has hyperbolic volume zero). Kill otherwise.
#  K3  mirror rule: the mirror replaces q by 1/q, which conjugates both invariants. 4_1 is amphichiral,
#      so its invariant must be real (imaginary part below 1e-9 relative); the trefoil is chiral and its
#      invariant is not real. Signatures: sigma(4_1) = 0, sigma(3_1) = -2, sigma(mirror 3_1) = +2
#      (Seifert-matrix computation). Kill if any fails.
import numpy as np
ok=True
def report(name,cond,val=''):
    global ok
    print(('PASS ' if cond else 'FAIL ')+name, val); ok=ok and cond
VOL=2.029883212819307
def kashaev41(N, conj=False):
    q=np.exp(2j*np.pi/N*(-1 if conj else 1)); lp=0.0; terms=[0.0]
    for k in range(1,N):
        lp+=2*np.log(abs(1-q**k)); terms.append(lp)
    t=np.array(terms); m=t.max(); return m+np.log(np.sum(np.exp(t-m)))
prev=None; rows=[]
for N in [50,100,200,400,800]:
    lk=kashaev41(N); raw=2*np.pi/N*lk; cor=2*np.pi/N*(lk-1.5*np.log(N)+0.25*np.log(3)); rows.append(cor)
    print(f'   N={N:4d}: (2pi/N) log<4_1> = {raw:.6f}   corrected = {cor:.6f}   (Vol = {VOL:.6f})')
report('K1 volume conjecture, figure-eight', abs(rows[-1]-VOL)<1e-2 and all(abs(rows[i+1]-VOL)<abs(rows[i]-VOL) for i in range(len(rows)-1)), f'error at N=800: {abs(rows[-1]-VOL):.2e}')
import mpmath as mp
def kz(N, mirror=False):
    # the partial products |(q;q)_n| reach about exp(0.16 N) before the sum cancels down to N^{3/2},
    # so the sum is done at a precision that grows with N
    mp.mp.dps=int(40+0.1*N)
    q=mp.exp(2j*mp.pi/N*(-1 if mirror else 1)); s=0; p=mp.mpc(1)
    for n in range(N):
        s+=p; p*=(1-q**(n+1))
    return complex(s)
r={}
for N in [100,200,400,800]:
    F=kz(N); r[N]=abs(F)/N**1.5; print(f'   N={N:4d}: |F|/N^1.5 = {r[N]:.6f}   arg F = {np.angle(F)/np.pi:+.6f} pi')
report('K2 trefoil Kontsevich-Zagier growth N^{3/2}', abs(r[800]-r[400])<1e-3, f'{r[400]:.6f} -> {r[800]:.6f}')
# K3 mirror
N=200; q=np.exp(2j*np.pi/N)
k41=sum(np.prod([abs(1-q**j)**2 for j in range(1,k+1)]) for k in range(N))
k41c=sum(np.prod([(1-q**j)*(1-q**(-j)) for j in range(1,k+1)]) for k in range(N))
report('K3 4_1 invariant is real (amphichiral)', abs(k41c.imag)<1e-9*abs(k41c), f'Im/Re = {k41c.imag/k41c.real:.1e}')
F=kz(N); Fm=kz(N,True)
report('K3 trefoil: mirror conjugates, invariant not real', abs(Fm-np.conj(F))<1e-9*abs(F) and abs(F.imag)>1e-3*abs(F), f'F = {F:.4f}, mirror = {Fm:.4f}')
def signature(V):
    S=V+V.T; ev=np.linalg.eigvalsh(S); return int(np.sum(ev>0)-np.sum(ev<0))
V31=np.array([[-1,1],[0,-1]]); V41=np.array([[1,1],[0,-1]])
s31=signature(V31); s31m=signature(-V31.T); s41=signature(V41); s41m=signature(-V41.T)
report('K3 signatures', s41==0 and s41m==0 and s31==-2 and s31m==2, f'sigma(4_1)={s41}, mirror {s41m}; sigma(3_1)={s31}, mirror {s31m}')
print('ALL PASS' if ok else 'SOME FAILED')