Knots, Graphs and Surfaces

Abigail Thompson (UCDavis)


Any connected, finite graph can be imbedded in the 3-dimensional sphere so that its complement has free fundamental group. Some graphs can be imbedded so that the complement of every subgraph also has free fundamental group. Planar graphs have such imbeddings, and so do some non-planar graphs, such as the complete graph on 5 vertices. However some graphs, such as the complete graph on 7 vertices, $K_7$, are intrinsically knotted; i.e., no matter how $K_7$ is imbedded in the 3-sphere, it contains a knotted cycle. I’ll talk about an integer invariant that measures how intrinsically knotted a graph is, and relate this invariant to a long-standing conjecture in graph theory, called the orientable cycle double cover conjecture.

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