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Wednesday, July 13, 2005

Cameron Gordon

Photo: Cameron Gordon (my Ph. D. advisor) with about half of his students at a conference for his 60'th birthday. I am the one with long hair, a gray beard, and a black t-shirt with planets on it.

Cameron Gordon is one of knot theory's top all time researchers. He is probably best known the Gordon-Lueke result (John Lueke was one of his top students and is the blonde guy behind me and to the left) which states that two knots are equivalent if and only if their complements are homeomorphic.

To the non-specialist: a knot will be considered as a smooth (smooth as meaning "differentiable") closed loop in the three sphere (which we consider as the standard 3 space with "infinity" considered to be a point; to visualize this, go one dimension down. Look at the plane and imagine what would happen if "infinity" was declared to be a point. That point could be, say, the north pole). Because of the differentiability condition, one can think of a knot as something like a shoestring whose ends have been fused.

A knot complement is what is left when the shoestring is removed from three space. So the Gordon-Lueke result says that if two knot complements have a one to one continuous function between them that has a continuous inverse, then the knots themselves must have been the "same knot".

Note: this result is well known to be FALSE for "links" (systems of more than one closed loop).


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