Incompressible punctured tori in the complements of alternating knots (Q1345932)

The alternating knots which allow an incompressible, boundary incompressible, punctured torus to be properly imbedded in their complements are determined to be the 2-bridge knots which have a continued fraction expansion of length two and the pretzel knots. All of the incompressible tori can be explicitly seen. For 2-bridge knots, the manifolds obtained by performing Dehn surgery on the boundary slopes of these punctured tori are shown to break up along incompressible tori into Seifert-fibred spaces.