Standard graphs in lens spaces

Ying-Qing Wu

Vol. 220 (2005), No. 2, 389–397

DOI: 10.2140/pjm.2005.220.389

Abstract

We prove the conjecture of Menasco and Zhang that a completely tubing compressible tangle consists of at most two families of parallel strands. This conjecture is related to problems concerning graphs in 3-manifolds, and follows from a theorem that states that a 1-vertex graph in M is standard, in a certain sense, if and only if the exteriors of all its nontrivial subgraphs are handlebodies.

Keywords

lens spaces, graphs, handlebodies

Mathematical Subject Classification 2000

Primary: 57N10

Milestones

Received: 4 February 2002

Revised: 6 January 2003

Accepted: 3 February 2003

Published: 1 June 2005