Abelianized peripheral data do not distinguish the unknot from the trefoil

Given: The crossing-free unknot diagram, the standard three-crossing trefoil, their compact exteriors and peripheral curves.

Full AC is the assertion that every set-indexed family of nonempty sets has a choice function. The Axiom of Choice

For any chosen complement basepoint and the diagram's common upper-collar access paths, the Wirtinger presentation uses one positive-meridian generator for each under-arc and one crossing conjugacy relation for each crossing, and identifies the presented group with the based complement fundamental group. Correctness of the Wirtinger presentation

The trefoil group based at the upper-collar basepoint specified in the trefoil computation is nonabelian and has abelianization $\mathbb{Z}$. The trefoil group is nonabelian with abelianization Z

Under countable choice, for every smooth knot $H_1(E_K;\mathbb{Z})\cong \mathbb{Z}$ is generated by $[\mu ]$, and the preferred longitude satisfies $[\lambda ]=0$. Homology and preferred longitude of a knot exterior

Ambient equivalence is given by an ambient isotopy of $S^3$, whose final diffeomorphism restricts to a homeomorphism of knot complements. Oriented links in the three-sphere and ambient isotopy

Based loop classes form a group under concatenation. A homeomorphism sends loops and endpoint-fixed homotopies to loops and endpoint-fixed homotopies, preserves concatenation, and its inverse induces the inverse group map. Based loops and the fundamental group Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation

Full AC [A1] supplies the countable-choice hypothesis of [F3] by restricting its choice function to each countable family of nonempty sets. The Axiom of Choice