Knotted handle decomposing spheres for handlebody-knots

Abstract

We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots 6₁₄ and 6₁₅ are not equivalent. We also show that certain genus two handlebody-knots with a knotted handle decomposing sphere can be determined by their exteriors. As an application, we show that the exteriors of 6₁₄ and 6₁₅ are not homeomorphic.