According to Sir Michael Atiyah [At], the study of topological quantum field theory is equivalent to the study of invariant quantities associated to three-dimensional manifolds. Although one has long considered the classical homology and cohomology structures and their extremely successful generalizations, the real subject of the Atiyah assertion is the new invariants proposed by Witten associated to the Jones polynomials of classical knots and links in the three-dimensional sphere. There have been many manifestations described by Reshetikhin & Turaev [Re1&2], Turaev & Viro [TV], Lickorish [Li 11– 15]. Kirby & Melvin [KM1&2], and Blanchet, Habegger, Mausbaum & Vogel [BHMV]. In these notes I describe some of the fundamental aspects of this theory, discuss the interest in these invariants and their extensions to the class of spatial graphs by Jonish & Millett [JonM], Kauffman & Vogel [KauV], Yamada [Ya2], Millett [Mi1&2], Kuperberg [Ku1&2], and Jaeger, Vertigan and Welsh [JaVW].
The fermion-boson mapping in three-dimensional quantum field theory
