Examples of tunnel number one knots which have the property ‘1 + 1 = 3’

Abstract

Let K be a knot in the 3-sphere S 3 , N ( K ) the regular neighbourhood of K and E ( K ) = cl(S 3 − N ( K )) the exterior of K . The tunnel number t ( K ) is the minimum number of mutually disjoint arcs properly embedded in E ( K ) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K . In particular, we call it an unknotting tunnel if the system consists of a single arc.