Local detection of strongly irreducible Heegaard splittings via knot exteriors. (Q1426491)

A Heegaard splitting \(H_1\bigcup_{\Sigma}H_2\) of a closed 3-manifold \(M\) is said to be weakly reducible if there exist meridian disks \(D_1,D_2\) of the handlebodies \(H_1,H_2\) such that \(\partial D_1\cap\partial D_2=\emptyset\). A Heegaard splitting which is not weakly reducible is called strongly irreducible. The authors study the intersection of a strongly irreducible Heegaard surface \(\Sigma\) with a submanifold \(X\) of \(M\) which is a cube-with-knotted-hole (i.e., such that (i) \(X\) is homeomorphic to the exterior of a non-trivial knot in \(S^3\) and (ii) there is a compressing disk \(D\) for \(\partial X\) whose boundary \(\partial D\) is a meridian curve of \(X\)). The main result of the paper is the following. If \(\Sigma \cap \partial X\) is a (non-empty) collection of simple closed curves which are essential in both \(\Sigma\) and \(\partial X\), then each component of \(\Sigma \cap X\) is a (possibly boundary parallel) meridional annulus.