- #1
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Hello,
Let's start with a cellular decomposition of 3-space, a "foam". A foam can be represented by its dual graph: cells and faces are dual to vertices and links.
What about the opposite?
It is clear that one can construct graphs for which no dual foam exists: take a large foam and connect two far-distant, inner vertices with a new link; as this link crosses many cells of the original foam the graph has no dual cellular decomposition in 3-space.
Is there a theorem which says something about higher dimensions?
One can emdedd any graph in 3-space, but for the dual foam this does no longer work. Is there a result showing that the dual foam would live in higher dimensional space? Is anything known about the required number of dimensions for the dual foam? Does it always exist?
Let's start with a cellular decomposition of 3-space, a "foam". A foam can be represented by its dual graph: cells and faces are dual to vertices and links.
What about the opposite?
It is clear that one can construct graphs for which no dual foam exists: take a large foam and connect two far-distant, inner vertices with a new link; as this link crosses many cells of the original foam the graph has no dual cellular decomposition in 3-space.
Is there a theorem which says something about higher dimensions?
One can emdedd any graph in 3-space, but for the dual foam this does no longer work. Is there a result showing that the dual foam would live in higher dimensional space? Is anything known about the required number of dimensions for the dual foam? Does it always exist?