Knots -- Eric Patterson, October 15, 2002

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Knot theory tries to understand the complexity of different embeddings of the 
circle in three dimensional Euclidean space and determine when two imbeddings 
are isotopic. Historically, knots were classified by their complexity as measured 
by their crossing number. For a given embedding, the crossing number can be 
quite difficult to determine. More recently, a large class of knots have 
been shown to correspond uniquely with 3-manifolds. The complexity of these 
manifolds can then be used as an alternative classification system. Computer 
programs make it easy to determine the relative complexity between two knots 
using this system; crossing number does not have this computational ease 
(at least no one has written such a program). This talk will introduce some 
basics of knots and 3-manifolds, how the two are related, and the progress 
made in producing a census of knots using this classification system.