Completely Tubing Compressible Tangles and Standard Graphs in Genus One 3-manifolds

Ying-Qing Wu Abstra t. 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 3manifold 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.