March 4, 2011 Alfredo Hubard, CIMS
Title: Space crossing numbers
If a graph with many edges is drawn in the plane then some of the edges have
to intersect. On the other hand, (by dimensional considerations) any graph
can be drawn in three dimensional space without two edges crossing.
Back to dimension two, mathematicians have wondered about the best way to
draw a graph in the plane, there are many answers to this question depending
on the context, one of particular interest is the crossing number. What is
the smallest number of crossings of any drawing of a given graph? There is a
rich theory of crossing numbers with interesting relations to engineering of
circuits and to geometric measure theory.
Our motivating question was: what is the analogue of the crossing number for
embeddings in three dimensions? Our answer (almost) recovers the most famous
result about (two dimensional) crossing numbers.