R3: quandle (x<|x = x): True  involutive: False
H_A  (n=0..12): [1, 3, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
H_B  (n=0..6) : [1, 3, 4, 3, 1, 0, 0]  (Fomin-Kirillov FK_3: 1,3,4,3,1)
H_K  (n=0..6) : [1, 3, 4, 3, 1, 0, 0]  (quadratic dual A^!)
H_A matches 1+3t+5t^2+6t^3/(1-t) up to t^12: True
H_A(t) H_B(-t) coefficients (t^0..t^12): [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0]
  = 1 - t^6 exactly (all other coefficients 0): True
Direct homology of Im f = A (x) FK_3 (x) A:
  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
  weight D=5: dim C_n (n=0..D) = [108, 219, 168, 57, 6, 0], ranks = [6, 102, 117, 51, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=6: dim C_n (n=0..D) = [144, 324, 292, 126, 19, 0, 0], ranks = [6, 138, 186, 106, 19, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 1}
  weight D=7: dim C_n (n=0..D) = [180, 432, 432, 219, 42, 0, 0, 0], ranks = [6, 174, 258, 174, 42, 0, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 3}
