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.
02/07/26
Submitted.
Last updated: 14:46 (GMT+2), 02/07/2026.