This project involves the completion, extension and further development of a purely combinatorial and axiomatic description of new three-dimensional invariants obtained from the Jones polynomial. These invariants were first described using ideas from quantum physics by the mathematical physicist E. Witten and later established by Reshethikin and Turaev, using the combinatorics of representations of quantum groups. A purely combinatorial approach was pioneered by Lickorish in the case of the classical one-variable Jones polynomial, evaluated at roots of unity of the parameter. This approach was recently extended by Blanchet-Habegger-Masbaum-Vogel to half-integer roots. They have also given a proof that the combinatorial theory satisfies the axioms for a topological quantum field theory first enunciated by Atiyah and Segal. The proof of the axioms is purely combinatorial using only the algebra and combinatorics of the Kauffman (skein) module of three-manifolds. The investigation of the axiomatics/combinatorics of the other Jones-Witten invariants may possibly be achieved using more general skein modules. Further, the rich supply of invariants should provide a basis for new approaches to old problems, such as link and three-manifold classification. Finally, the combinatorial approach will likely lead to effective methods of computation. Knots are rather elementary geometric objects whose really interesting properties are topological. By this we mean that two geometric knots do not differ in an interesting way if one of them can be transformed to look just like the other without cutting or untying it, just by pushing its string about to rearrange the crossings. Topologists say that they are two different geometric realizations of the same topological knot. Nevertheless, it is not a trivial matter to recognize when one complicated geometric knot is topologically different from another, rather than just a different geometric realization. This problem can be addressed by computing certain numbers or polynomials which are called "topological invariants," meaning that they always have the same value for different geometric realizations of the same topological knot. The problem would be reduced to pure algebra if there were one invariant which also always has different values for geometric realizations of different topological knots, but life is not so simple--no single invariant achieves this ideal, nor even all the known invariants taken together. It is therefore valuable to investigate new invariants, some of the most useful being those inspired in recent years by ideas from quantum physics. In particular, applications of knot theory to the biology of long strands of DNA have drawn upon knowledge of these newer invariants.
