We give a new, diagrammatic, proof, involving virtual knots, that if the quotient of the fundamental group of a blackboard-framed, oriented knot by the normal closure of its longitude is trivial, then it is a ±1-framed unknot. The arguments make use only of Reidemeister moves (no Kirby moves), providing new proofs of several important theorems concerning the 3-sphere.
24/09/26
Submitted (27/06/26). I continue to revise the article when I feel the exposition can be improved; the ideas and the line of argument are unchanged from the submitted version.
Last updated: 12:25 (GMT+2), 24/09/2026.