File size: 3,770 Bytes
848d4b7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | from mpmath import mp
mp.dps = 110
def bloch_wigner(z):
# D(z) = Im(Li_2(z)) + Arg(1-z)*log|z|
return mp.im(mp.polylog(2, z) + mp.log(1 - z) * mp.log(abs(z)))
def compute():
# Hyperbolic volume of the 7_2 knot complement.
# The 7_2 knot is a twist knot (two-bridge knot K(11,5)).
#
# Approach: Solve the gluing equations of the ideal triangulation obtained
# from SnapPy (4 tetrahedra, triangulation code "evQkbccddtnrnj_BbDc").
# Starting from SnapPy's 60-digit shape parameters, refine to 110+ digits
# via Newton's method on the log-form gluing equations.
#
# Gluing equations from SnapPy (format: A_vec, B_vec, sign):
# Eq 0: ([1,2,0,0], [-1,0,1,0], -1)
# Eq 1: ([0,-1,1,-2], [-1,1,0,2], -1)
# Eq 2: ([0,-1,-1,1], [1,-1,0,0], -1)
# Eq 3: ([-1,0,0,1], [1,0,-1,-2], -1)
# Eq 4: ([0,-1,0,0], [0,0,-1,0], 1) # meridian
#
# We use equations 0,1,2,4 (3 independent edge + 1 cusp completeness).
with mp.extradps(30):
# Starting shape parameters from SnapPy high_precision (60 digits)
z = [
mp.mpc(
"0.979683927137063080360443583225912498526944739792254472909696",
"0.590569559841547738085433207813503541833670692235462901341630",
),
mp.mpc(
"0.251322701057396787068916574052517527698543073419837511877978",
"0.451314970729364036154899986170441362413612486336944204016703",
),
mp.mpc(
"0.05818137738476620957186092260681916651032819794670750704818",
"1.69127914951419451109509131997221641885831120673024304031914",
),
mp.mpc(
"1.16369117147491476375354246222499900315270704909808869777148",
"0.56418563226878988033974884693917445186365596844491528772036",
),
]
# Gluing equation exponents (using equations 0,1,2,4)
A = [
[1, 2, 0, 0],
[0, -1, 1, -2],
[0, -1, -1, 1],
[0, -1, 0, 0],
]
B = [
[-1, 0, 1, 0],
[-1, 1, 0, 2],
[1, -1, 0, 0],
[0, 0, -1, 0],
]
signs = [-1, -1, -1, 1]
# Determine target values from approximate solution
targets = []
for i in range(4):
val = sum(A[i][j] * mp.log(z[j]) + B[i][j] * mp.log(1 - z[j])
for j in range(4))
# Round to nearest multiple of pi*i
k = round(float(mp.im(val) / mp.pi))
targets.append(mp.mpc(0, k * mp.pi))
# Newton's method to refine shapes to full precision
for iteration in range(10):
# Evaluate residuals
g = []
for i in range(4):
val = sum(A[i][j] * mp.log(z[j]) + B[i][j] * mp.log(1 - z[j])
for j in range(4))
g.append(val - targets[i])
# Check convergence
max_err = max(abs(gi) for gi in g)
if max_err < mp.mpf(10) ** (-(mp.dps + 20)):
break
# Compute Jacobian (4x4 complex matrix)
J = mp.matrix(4, 4)
for i in range(4):
for j in range(4):
J[i, j] = A[i][j] / z[j] - B[i][j] / (1 - z[j])
# Solve J * dz = -g
g_vec = mp.matrix([g[0], g[1], g[2], g[3]])
dz = mp.lu_solve(J, -g_vec)
# Update shape parameters
for j in range(4):
z[j] += dz[j]
# Compute volume as sum of Bloch-Wigner values
vol = sum(bloch_wigner(zi) for zi in z)
return mp.re(vol)
if __name__ == "__main__":
print(str(compute()))
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