Invariants of Links and 3-Manifolds, Their Properties and Topology
Thang T. Q. Le Principal Investigator
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Proposal: DMS-9971350<br/><br/>PI: Thang Le<br/><br/>Title: "Invariants of knots and 3-manifolds, their properties and topology"<br/><br/>Abstract: This research studies quantum and finite type invariants of<br/>3-manifolds and their relationships with classical topological and<br/>geometrical invariants. The investigator will continue to study<br/>the universal finite type invariant of homology 3-spheres which he<br/>developed in joint work with J. Murakami and T. Ohtsuki. In<br/>particular, he plans to develop the theory further to include<br/>other 3-manifolds and to study the topological quantum field<br/>theory associated with the Le-Murakmi-Ohtsuki invariant which is<br/>different from those associated with usual quantum invariants. He<br/>will try to understand the topology of these new invariants and<br/>find their applications. One of the projects is to study relations<br/>between the hyperbolic, or more general, simplicial, volume of<br/>knots and their quantum invariants (to prove the<br/>Kashaev-Murakami-Murakami conjecture).<br/><br/>The theory of knots and 3-manifolds is an old branch of<br/>mathematics which has gained renewed interest among mathematicians<br/>and physicists after the discovery of the Jones polynomial and its<br/>relation to theoretical physics (quantum field theory, high energy<br/>physics). In fact, it is now one of the most active domains in<br/>mathematics. Many results of knot theory may also find<br/>applications in molecular biology. To classify knots and<br/>3-manifolds, mathematicians use "invariants". This research<br/>project studies new classes of invariants of knots and 3-manifolds<br/>and their relationships with the classical ones. The new<br/>invariants are very powerful in distinguishing knots and<br/>3-manifolds.