Completely tubing compressible tangles and standard graphs in genus one 3-manifolds

Abstract

We prove a conjecture of Menasco and Zhang that if a tangle is completely tubing compressible then it consists of at most two families of parallel strands. This is related to problems of graphs in 3-manifold. A 1-vertex graph $Γ$ in a 3-manifold $M$ with a genus 1 Heegaard splitting is standard if it consists of one or two parallel sets of core curves lying in the Heegaard splitting solid tori of $M$ in the standard way. The above conjecture then follows from the theorem which says that a 1-vertex graph in $M$ is standard if and only if the exteriors of all its nontrivial subgraphs are handlebodies.