In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a
graph which we call "The Big Dehn Surgery Graph", B. Here we explore this
graph, particularly the link of S^3, and prove facts about the geometry and
topology of B. We also investigate some interesting subgraphs and pose what we
believe are important questions about B.Comment: 15 pages, 4 figures, 4 ancillary files. Reorganized and shortened
from previous versions, while correcting one error in the proof of Theorem
5.4. Also, ancillary files detailing our computations with the computer
program ORB have been provide
