This project concerns applications of techniques from classical 3-dimensional topology and hyperbolic geometry to the study of polynomial link invariants and finite type invariants of links and 3-manifolds: (a) The PI has proved results suggesting deep connections between the volume of hyperbolic links and the coefficients of their Jones polynomial. She would like to further investigate these connections and establish a bridge between quantum topology and hyperbolic geometry. (b) Investigate the connections of link polynomial invariants to a graph theoretic polynomials (e.g. the Bollob'as ?Riordan polynomial). She hopes that studying link invariants within this framework, can lead to new connections between link polynomials and Khovanov homology and geometric structures of link complements. (c) Use the general machinery developed to understand how 3-manifolds change under surgery to explore the topological relations captured by the finite type knot invariants. The main tools here include sutured 3-manifold theory, combinatorial techniques from Dehn surgery and surface mapping group techniques.
The research of the project lies in the area of 3-dimensional topology the central objects of study of which 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. One of the ways that topologists have been approaching these problems is through the use of invariants. In the recent years, ideas originated in physics, lead mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. The central theme of the PI's project is to understand the properties of these invariants, using ideas from 3-dimensional topology and geometry and from physics, and investigate the extent to which they distinguish knots and 3-manifolds.
