Standard graphs in lens spaces (Q2496560)

A pair \((W,t)\) is called a tangle if \(W\) is a compact orientable three-manifold with boundary \(\partial{W}\) a sphere, and \(t:=t_{1}\cup t_{2}\cup\dots\cup t_{k}\) is a set of properly embedded arcs in \(W\). Put \(X:=W\setminus\operatorname {Int}N(t)\), \(A_j:=\partial{N(t_j)}\cap{X}\), and \(F_j:=(\partial{W}\cap{X})\cup A_{j}\), where \(N(t_{j})\) is the tubular neighborhood of \(t_{j}\). A tangle \((W,t)\) is called completely tubing compressible if \(F_{i}\) can be compressed until it becomes a set of annuli parallel to \(\bigcup_{j\neq i}A_{j}\) for every \(i\). In the paper under review the author proves the following theorem: if a tangle \((W,t)\) is completely tubing compressible, then \(t\) consists of at most two families of parallel core arcs. Here an arc in \(W\) is called a core arc if \(W\setminus\operatorname{Int}N(\alpha)\) is a solid torus. This is proved by using the fact that a tangle \((W,t)\) is completely tubing compressible if and only if the exterior of any non-trivial subgraph of \(\widehat{t}\) in \(\widehat{W}\) is a handlebody, where \(\widehat{W}\) is a closed three-manifold obtained from \(W\) by adding a three-ball and \(\widehat{t}\subset\widehat{W}\) is a graph obtained from \(t\) by adding a vertex in the three-ball. The theorem was conjectured by \textit{W.~Menasco} and \textit{X.~Zhang} [Pac. J. Math. 198, No. 1, 149--174 (2001; Zbl 1049.57013)].