The article "The Combinatorial Revolution in Knot Theory" by Sam Nelson was featured in the Notices of the American Mathematical Society in its December 2011 issue. Sam Nelson received his PhD from LSU in 2002 under Professor Rick Litherland. He is currently Assistant Professor at Claremont McKenna College, CA.
The AMS writes in its news release:
"In his article ... Sam Nelson describes a novel approach to knot theory that has gained currency in the past several years and the mysterious new knot-like objects discovered in the process ...
Just as new horizons opened when people dared to consider what would happen if -1 had a square root---and thereby discovered complex numbers, which have since been thoroughly explored by mathematicians and have become ubiquitous in physics and engineering---mathematicians are finding that the equations they used to investigate regular knots give rise to a whole universe of "generalized knots" that have their own peculiar qualities. Although they seem esoteric at first, these generalized knots turn out to have interpretations as familiar objects in mathematics. "Moreover," Nelson writes, "classical knot theory emerges as a special case of the new generalized knot theory." "