== Z/4 permutation rack sigma(x,y) = (y, x+1)
properties: {'ybe': True, 'bijective': True, 'left_nondeg': True, 'right_nondeg': True, 'involutive': False, 'idempotent': False}
both sides of the braid relation give (z, y+1, x+2): True
sigma has no fixed points: True
sigma^2(x,y) = (x+1, y+1): True
dim A_n (monoid classes), n=0..8: [1, 4, 2, 1, 1, 1, 1, 1, 1]
A_2 classes: P and Q as described: True
u0 a = {'P': 0, 'Q': 0}  u0 b = {'P': 0, 'Q': 0}
aa = {'P': 2, 'Q': -2}  bb = {'P': 2, 'Q': -2}  ab+ba = {'P': 0, 'Q': 0}
dim B_n (exact, X-basis), n=0..4: [1, 4, 14, 47, 152]
dim K_n (exact, X-basis), n=0..5: [1, 4, 14, 49, 171, 597]
B_n in K_n (stack o QS_n == 0) for n=2..4: True
H_A(t) H_B(-t) up to t^4: [1, 0, 0, 2, -11]
H_A(t) H_K(-t) up to t^5: [1, 0, 0, 0, 0, 0]
weight 1: dims C_0..C_1 = [8, 4], dim A_w = 4; H_-1(aug)=0, H_n = [0, 0]
weight 2: dims C_0..C_2 = [20, 32, 14], dim A_w = 2; H_-1(aug)=0, H_n = [0, 0, 0]
weight 3: dims C_0..C_3 = [18, 80, 112, 47], dim A_w = 1; H_-1(aug)=0, H_n = [0, 0, 2, 0]
weight 4: dims C_0..C_4 = [14, 72, 280, 376, 152], dim A_w = 1; H_-1(aug)=0, H_n = [0, 0, 0, 3, 0]
over Q: middle factors of z_re, z_im lie in R = B_2: True
over Q: d z_re = 0: True  d z_im = 0: True
over Q: rank of weight-3 boundaries in C_2 = 47, with z_re, z_im adjoined = 49 -> independent modulo boundaries: True
-- positive characteristic (complex over F_p; exact ranks mod p)
p=3: dim B_2=14, dim B_3=46, dim K_3=49; middles in R: True; rank bdry=46, with z's=48; z_re,z_im independent mod boundaries: True
p=5: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=7: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=11: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=13: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=17: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=19: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=23: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=29: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=31: dim B_2=14, dim B_3=47, dim K_3=49; middles in R: True; rank bdry=47, with z's=49; z_re,z_im independent mod boundaries: True
p=2: dim B_2=14, dim B_3=46, dim K_3=49; middles in R: True; rank bdry=46, with z's=46; z_re,z_im independent mod boundaries: False
Thm 1.3(c): dim K_3 = 49 , dim B_3 = 47
Remark 3.3: H_n(B,partial), n=0..3: [1, 1, 0, 1]
Remark 3.3: H_n(K,partial), n=0..3: [1, 1, 0, 0]
time 1.2s
