Alexander--Groebner computation
================================
name: Hopf
invariant: AG_1
coefficient domain: QQ
term order: lex: t1 > t2

Definition used by this program:
  n = 2 presentation generators
  E_1 = I_{n-1}(A) = I_1(A),
  where A is the abelianized Fox matrix and
  AG_k is the reduced lexicographic Groebner basis of
  E_k intersected with QQ[t1,t2].

Fox matrix shape: 1 x 2
minor size: 1
candidate minors: 2
nonzero minors: 2
distinct polynomial minor generators: 2

AG_1 (factored for display):
{
  t1 - 1,
  t2 - 1
}
normalized common divisor: 1

Internal checks:
  PASS  fox_fundamental_identity
  PASS  determinantal_generators_in_reported_ideal
  PASS  basis_stable_under_reduced_lex_recomputation
  PASS  independent_reducedness_and_buchberger_check

elapsed time: 0.023572 s
mathematical-input SHA-256: e18032d97da26e488e364c414bb854cfbe14a176315734104cbffb6da87a0889
normalized-input-record SHA-256: 74764499a9d38886ca5143685184e55a147ffe7a2478a471eeb51a958d8fa36b
exact source-JSON SHA-256: e6c552777a41329a467e49a85771c6c0d6f2350927ae507faf3d47244ed395de
