Tunnel Number One Genus One Non-Simple Knots

Abstract

A knot in the 3-sphere is said to be tunnel number one if there exists an arc attached to the knot at its endpoints so that the complement of a regular neighbourhood of the resulting complex is a genus two handlebody. It is well-known that every torus knot and 2-bridge knot is tunnel number one. Although no composite knot is tunnel number one $[7, 8]$ , some non-simple knots are known to be tunnel number one It seems difficult to characterize tunnel number one knots.