Abstract
In a talk at the Cornell Topology Festival in 2004, W. Thurston discussed a graph which we call “The Big Dehn Surgery Graph”, B \mathcal {B} . Here we explore this graph, particularly the link of S 3 S^3 , and prove facts about the geometry and topology of B \mathcal {B} . We also investigate some interesting subgraphs and pose what we believe are important questions about B \mathcal {B} .
Highlights
There is a knot K ⊂ M and M is obtained by non-trivial Dehn surgery along K in M
The edges are unoriented since M is obtained from M via Dehn surgery
We first record some basic properties of B. These follow from just some of the extensive work that has been done in the field of Dehn surgery
Summary
We first record some basic properties of B These follow from just some of the extensive work that has been done in the field of Dehn surgery. The graph B is connected by the beautiful work of Lickorish [21] and Wallace [35] who independently showed that all closed, orientable 3-manifolds can be obtained by surgery along a link in S3. This shows that a characterization of the vertices in the link of S3 remains open.
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