On unknotting tunnels for knots
The main purpose of this paper is to determine the non-simple knots in \(S^ 3\) with tunnel number one, and to classify the unknotting tunnels for them. Roughly speaking, such knots are certain satellites of torus knots via 2-bridge links; and most of them admit precisely 4 (infinitely many resp.) unknotting tunnels up to homeomorphism (isotopy resp.). To understand the unknotting tunnels for them, we introduce concepts of a dual tunnel and a (1,1)-tunnel, and we also review the unknotting tunnels for torus knots and 2-bridge knots from this view point. To detect inequivalences of the unknotting tunnels for non-simple knots, we use a very natural homotopy invariant; and we also classify the (known) unknotting tunnels for torus knots and 2-bridge knots through the invariant.
non-simple knots in \(S^ 3\) with tunnel number one
satellites of torus knots via 2-bridge links
scientific article; zbMATH DE number 4143045
scientific article