Rectangle condition for compression bodies and 2-fold branched coverings (Q2885409)

The importance of strong irreducibility for Heegaard splittings of a 3-manifold is well-known; for example, a 3-manifold admitting a strongly irreducible Heegaard splitting is irreducible and boundary-irreducible. In an unpublished paper, Casson and Gordon defined the so-called \textit{rectangle condition} for Heegaard diagrams of closed 3-manifolds, which implies that the given splitting is strongly irreducible, see \textit{T. Kobayashi}, [Osaka J. Math. 25, No. 3, 553--573 (1988; Zbl 0709.57009)].NEWLINENEWLINEIn the present paper, a \textit{rectangle condition} for Heegaard splittings of 3-manifolds with non-empty boundary is defined; as in the closed case, it is proved to imply strong irreducibility of Heegaard splittings.NEWLINENEWLINEThen, the authors introduce a notion of \textit{generalized Heegaard splittings} (relying on a decomposition of the manifold via a collection of surfaces) and both a \textit{thin position} and a \textit{rectangle condition} for knots and links. By applying the rectangle condition to a generalized Heegaard splitting of a 2-fold branched covering of the 3-sphere, and by making use of the result by \textit{M. Scharlemann} and \textit{A. Thomson} [Contemp. Math. 164, 231--238 (1994; Zbl 0818.57013)], the rectangle condition is proved to imply that any thin surface in the 2-fold covering is incompressible.NEWLINENEWLINEFinally, the rectangle condition is applied to the additivity of knot width: if a composite knot in thin position satisfies the rectangle condition, then the width of the knot is additive with respect to connected sum.