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A Chapter in Physical Mathematics: Theory of Knots in the Sciences
A systematic study of knots was begun in the second half of the 19th century by Tait and his followers. They were motivated by Kelvin's theory of atoms modelled on knotted vortex tubes of ether. It was expected that physical and chemical properties of various atoms could be expressed in terms of properties of knots such as the knot invariants. Even though Kelvin's theory did not work, the theory of knots grew as a subfield of combinatorial topology. Recently new invariants of knots have been discovered and they have led to the solution of long standing problems in knot theory. Surprising connections between the theory of knots and statistical mechanics, quantum groups and quantum field theory are emerging. We give a geometric formulation of some of these invariants using ideas from topological quantum field theory. We also discuss some recent connections and application of knot theory to problems in Physics, Chemistry and Biology. It is interesting to note that as we stand on the threshold of the new millenium, difficult questions arising in the sciences continue to serve as a driving force for the development of new mathematical tools needed to understand and answer them.