An invariant derived from the Alexander polynomial for handlebody-knots

Abstract

A handlebody-knot is a handlebody embedded in the $3$-sphere. We introduce an invariant for handlebody-knots derived from their Alexander polynomials. The value of the invariant is a vertex-weighted graph. As an application, we describe a sufficient condition for a handlebody-knot to be irreducible and a necessary condition for a link to be a constituent link of a handlebody-knot.