2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/225418In this paper, we introduce a notion called n/k-free tangle and study the degeneration ratio of tunnel numbers of knots.This is the version published by Geometry & Topology Monographs on 3 December 2007Geometric Topology57M25, 57N10text