The project addresses relations between geometry, topology and combinatorial invariants of 3-manifolds. The PI will establish concrete connections between geometry and topological descriptions, properties, and quantum invariants of links and 3-manifolds. One part of the project will combine several techniques, developed both by the PI and her collaborators and by others, to derive estimates of hyperbolic volumes and other geometric invariants of 3-manifolds from topological data such as Dehn surgery presentations, link diagrams and group actions. A second part will establish relations between the Jones knot polynomials (and the Khovanov homology), essential surfaces, and geometric structures of knot complements. A third part will study skein link theory in 3-manifolds, its invariants, and investigate its interaction with the detailed structures coming from the geometrization picture. The research also involves graduate students currently working with PI.
The research of the project lies in the area of 3-dimensional topology. The central objects of this study are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3- dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. The solution of Thurston's Geometrization Conjecture has established that 3-manifolds (and complements of knots in them) decompose into pieces that admit explicit geometries and that hyperbolic geometry is the one that appears more often. In practice, however, 3-manifolds are often given in terms of combinatorial topological descriptions and it is both natural and important to seek for ways to deduce geometric information from these descriptions. One of the ways that topologists have been approaching the study of 3- manifolds is through the use of invariants. In the last few decades ideas originated in physics led mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. Understanding the connections of topological and combinatorial quantities and invariants to geometry is a central and important goal of 3-dimensional topology. The main theme of this project is to establish concrete such connections and explore their ramifications and applications to other areas of mathematics.