eq. (8): QS_3|W = (1-l)[[1,-1,1],[-l,1+l,-1],[l^2,-l,1]] as polynomials in l: True
det = (1-l)^4 (1+l^2): True  det coefficients: [(0, 1), (1, -4), (2, 7), (3, -8), (4, 7), (5, -4), (6, 1)]
QS_3(u1^3) coefficient (1+q)(1+q+q^2), q=-omega: -2-4w ; equals -2 omega (1-omega): True
QS_3(u1u1u1) from the model: {(1, 1, 1): '-2-4w'}
lambda=omega: det of the 3x3 block = -9+0w
lambda=omega: QS_4 on v1..v4 (rows v1..v4, column j = QS_4(v_j)):
    ['-2-4w', '2+4w', '-2-4w', '2+4w']
    ['-4-2w', '4-1w', '-1+4w', '-2-4w']
    ['-2+2w', '-4-5w', '4-1w', '2+4w']
    ['2+4w', '-2+2w', '-4-2w', '-2-4w']
equals the printed 4x4 matrix: True
l' vanishes on the columns: True ; l'(v4) = 1: True ; rank: 3
(ad_c u1)^3(u0) = ['1+0w', '2+2w', '0+2w', '-1+0w'] ; kernel dim: 1 ; spans the kernel: True
AS02 Lemma 3.7(b) factors for r=3: (3)!_q11 = -2-4w , 1-q10q01 = 1-1w , 1-q11q10q01 = 0-1w , 1-q11^2q10q01 = 0+0w
lambda=i, multidegree a0+2a1: chain dims {0: 0, 1: 3, 2: 6, 3: 2}  homology {-1: 0, 0: 0, 1: 0, 2: 1, 3: 0}
   z = u0(x)u1u1(x)1 - 1(x)u0u1(x)u1: (cycle, middles in B_2, not a boundary) = (True, True, True)
lambda=omega, multidegree a0+3a1: chain dims {0: 0, 1: 0, 2: 4, 3: 8, 4: 3}  homology {-1: 0, 0: 0, 1: 0, 2: 0, 3: 1, 4: 0}
   z' = u0(x)u1u1u1(x)1 + 1(x)u0u1u1(x)u1: (cycle, middles in B_3, not a boundary) = (True, True, True)
control lambda=-1: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=zeta6: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=zeta12: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=zeta5: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=zeta5^2: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=zeta7: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=zeta8: non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {})]
control lambda=i (again): non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {2: 1}), ((0, 1, 1, 1), {})]
control lambda=omega (again): non-zero homology in a0+2a1, a0+3a1: [((0, 1, 1), {}), ((0, 1, 1, 1), {3: 1})]
time 0.1s
