C-essential surfaces in (3-manifold, graph) pairs (Q357215)

There are several versions of thin position for knots and 3-manifolds, this paper treats a very general one. Let \(M\) be a compact orientable 3-manifold possibly with boundary, let \(T\) be a graph properly embedded in \(M\) and \(\Gamma\) a subgraph of \(T\) disjoint from the vertices of \(T-\partial T\). Suppose that \(M-T\) is irreducible and no sphere in \(M\) intersects \(T\) exactly once. The main result is somehow technical, but roughly speaking says the following: Suppose \(H\) is a Heegaard surface for \(M\), so that \(T\) is in bridge position with respect to \(H\). Then either a degenerate situation occurs or \(H\) can be untelescoped into a collection of thick and thin surfaces. The thin surfaces are \(c\)-essential in the graph exterior and each thick surface is a strongly irreducible bridge surface in the complement of the thin surfaces. In the case that \(M\) is a 3-manifold, \(T\) a properly embedded 1-manifold and \(\Gamma=\emptyset\), this produces a stronger version of thin position as defined by \textit{C. Hayashi} and \textit{K. Shimokawa} [Pac. J. Math. 197, No. 2, 301--324 (2001; Zbl 1050.57016)]. In the case that \(M\) is closed, \(T\) is a link and \(\Gamma=T\), this produces a stronger version of thin position as defined by \textit{M. Tomova} [J. Lond. Math. Soc., II. Ser. 80, No. 1, 85--98 (2009; Zbl 1220.57004)].