==========================================================================================
SOLUTION flip3 (involutive control)  N = 3  bijective True  YBE True
 (1) f o d = eps * b o f on all basis elements of weight <= 3: True (132, {1: {-1}, 2: {-1}, 3: {-1}})
 (2) inner bar terms vanish on B_n (mu_k QS_n = 0), n=2..4: True
 (3) homology of Im f = A (x) B (x) A (FGG Question 44 asks: acyclic?)
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
 (4a) homology of the full braided complex B (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [21, 18, 9], homology: {2: 6}
  weight D=3: dims = [56, 63, 54, 27], homology: {2: 18, 3: 8}
  weight D=4: dims = [126, 168, 189, 162, 81], homology: {2: 36, 3: 24, 4: 39}
 (4b) homology of the quadratic quotient B/<omega> (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [21, 18, 3], homology: ACYCLIC
  weight D=3: dims = [56, 63, 18, 1], homology: ACYCLIC
  weight D=4: dims = [126, 168, 63, 6, 0], homology: ACYCLIC
 time 0.0s
==========================================================================================
SOLUTION Lyubashenko3 (involutive control)  N = 3  bijective True  YBE True
 (1) f o d = eps * b o f on all basis elements of weight <= 3: True (132, {1: {-1}, 2: {-1}, 3: {-1}})
 (2) inner bar terms vanish on B_n (mu_k QS_n = 0), n=2..4: True
 (3) homology of Im f = A (x) B (x) A (FGG Question 44 asks: acyclic?)
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
 (4a) homology of the full braided complex B (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [21, 18, 9], homology: {2: 6}
  weight D=3: dims = [56, 63, 54, 27], homology: {2: 18, 3: 8}
  weight D=4: dims = [126, 168, 189, 162, 81], homology: {2: 36, 3: 24, 4: 39}
 (4b) homology of the quadratic quotient B/<omega> (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [21, 18, 3], homology: ACYCLIC
  weight D=3: dims = [56, 63, 18, 1], homology: ACYCLIC
  weight D=4: dims = [126, 168, 63, 6, 0], homology: ACYCLIC
 time 0.0s
==========================================================================================
SOLUTION dihedral quandle R3 (control, B=FK_3)  N = 3  bijective True  YBE True
 (1) f o d = eps * b o f on all basis elements of weight <= 3: True (132, {1: {-1}, 2: {-1}, 3: {-1}})
 (2) inner bar terms vanish on B_n (mu_k QS_n = 0), n=2..4: True
 (3) homology of Im f = A (x) B (x) A (FGG Question 44 asks: acyclic?)
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
 (4a) homology of the full braided complex B (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [19, 18, 9], homology: {2: 5}
  weight D=3: dims = [42, 57, 54, 27], homology: {2: 9, 3: 3}
  weight D=4: dims = [73, 126, 171, 162, 81], homology: {2: 11, 3: 5, 4: 25}
 (4b) homology of the quadratic quotient B/<omega> (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [19, 18, 4], homology: ACYCLIC
  weight D=3: dims = [42, 57, 24, 3], homology: ACYCLIC
  weight D=4: dims = [73, 126, 76, 18, 1], homology: ACYCLIC
 time 0.0s
==========================================================================================
SOLUTION permrack2_10  N = 2  bijective True  YBE True
 (1) f o d = eps * b o f on all basis elements of weight <= 3: True (50, {1: {-1}, 2: {-1}, 3: {-1}})
 (2) inner bar terms vanish on B_n (mu_k QS_n = 0), n=2..5: True
 (3) homology of Im f = A (x) B (x) A (FGG Question 44 asks: acyclic?)
  weight D=1: dim C_n (n=0..D) = [4, 2], ranks = [2, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [6, 8, 3], ranks = [1, 5, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [6, 12, 12, 5], ranks = [1, 5, 7, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [7, 12, 18, 20, 8], ranks = [1, 6, 6, 12, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=5: dim C_n (n=0..D) = [8, 14, 18, 30, 32, 13], ranks = [1, 7, 7, 11, 19, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
 (4a) homology of the full braided complex B (augmented)
  weight D=1: dims = [4, 2], homology: ACYCLIC
  weight D=2: dims = [6, 8, 4], homology: {2: 1}
  weight D=3: dims = [6, 12, 16, 8], homology: {2: 2, 3: 1}
  weight D=4: dims = [7, 12, 24, 32, 16], homology: {2: 1, 3: 1, 4: 2}
 (4b) homology of the quadratic quotient B/<omega> (augmented)
  weight D=1: dims = [4, 2], homology: ACYCLIC
  weight D=2: dims = [6, 8, 3], homology: ACYCLIC
  weight D=3: dims = [6, 12, 12, 5], homology: ACYCLIC
  weight D=4: dims = [7, 12, 18, 20, 8], homology: ACYCLIC
 time 0.0s
==========================================================================================
SOLUTION permrack4_1230  N = 4  bijective True  YBE True
 (1) f o d = eps * b o f on all basis elements of weight <= 3: True (324, {1: {-1}, 2: {-1}, 3: {-1}})
 (2) inner bar terms vanish on B_n (mu_k QS_n = 0), n=2..4: True
 (3) homology of Im f = A (x) B (x) A (FGG Question 44 asks: acyclic?)
  weight D=1: dim C_n (n=0..D) = [8, 4], ranks = [4, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [20, 32, 14], ranks = [2, 18, 14]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [18, 80, 112, 47], ranks = [1, 17, 63, 47]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {2: 2}
  weight D=4: dim C_n (n=0..D) = [14, 72, 280, 376, 152], ranks = [1, 13, 59, 221, 152]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 3}
 (4a) homology of the full braided complex B (augmented)
  weight D=1: dims = [8, 4], homology: ACYCLIC
  weight D=2: dims = [20, 32, 16], homology: {2: 2}
  weight D=3: dims = [18, 80, 128, 64], homology: {2: 6, 3: 5}
 (4b) homology of the quadratic quotient B/<omega> (augmented)
  weight D=1: dims = [8, 4], homology: ACYCLIC
  weight D=2: dims = [20, 32, 14], homology: ACYCLIC
  weight D=3: dims = [18, 80, 112, 49], homology: {2: 2, 3: 2}
  weight D=4: dims = [14, 72, 280, 392, 171], homology: {3: 5, 4: 5}
 time 0.3s
==========================================================================================
SOLUTION permrack3_120  N = 3  bijective True  YBE True
 (1) f o d = eps * b o f on all basis elements of weight <= 3: True (150, {1: {-1}, 2: {-1}, 3: {-1}})
 (2) inner bar terms vanish on B_n (mu_k QS_n = 0), n=2..5: True
 (3) homology of Im f = A (x) B (x) A (FGG Question 44 asks: acyclic?)
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [13, 18, 7], ranks = [2, 11, 7]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [14, 39, 42, 16], ranks = [1, 13, 26, 16]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [12, 42, 91, 96, 33], ranks = [1, 11, 31, 60, 33]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 3}
  weight D=5: dim C_n (n=0..D) = [12, 36, 98, 208, 198, 64], ranks = [1, 11, 25, 73, 134, 64]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 1}
 (4a) homology of the full braided complex B (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [13, 18, 9], homology: {2: 2}
  weight D=3: dims = [14, 39, 54, 27], homology: {2: 2, 3: 1}
  weight D=4: dims = [12, 42, 117, 162, 81], homology: {2: 1, 3: 4, 4: 8}
 (4b) homology of the quadratic quotient B/<omega> (augmented)
  weight D=1: dims = [6, 3], homology: ACYCLIC
  weight D=2: dims = [13, 18, 7], homology: ACYCLIC
  weight D=3: dims = [14, 39, 42, 16], homology: ACYCLIC
  weight D=4: dims = [12, 42, 91, 96, 36], homology: {3: 3, 4: 3}
 time 0.1s
