Geometric Aspects Knot and 3-manifold Invariants

Michigan State University, East Lansing MI

Investigators

Abstract

The research in this NSF funded project lies in the area of three-dimensional topology, where the central objects of study are spaces called three-manifolds.  A three-manifold is a space that locally looks like the ordinary three-dimensional space but whose global structure may be complicated. An important part of three-dimensional topology is also the study of knots, or in other words, loops embedded in some tangled way in three-manifolds. The solution of a well-known problem known as Thurston's Geometrization Conjecture has established that three-manifolds, and complements of knots in them, decompose into pieces that admit explicit geometries. One of the most common and most interesting geometries that appear in this setting is hyperbolic geometry.  In practice, three-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 three-manifolds is through the construction and study of objects called invariants. In the last few decades, ideas that originated in quantum physics have led mathematicians to the discovery of a variety of subtle and powerful invariants of knots and three-manifolds.  Understanding the connections of topological and combinatorial quantities and invariants to detailed geometric structures, arising from Thurston's picture, is a central and important goal of low dimensional topology.  The main theme of this project is to establish such connections and to explore their ramifications and applications to topology  as well as other areas of mathematics. The project aims to establish intrinsic connections between geometry and topological descriptions, properties, and quantum invariants of links and three-manifolds. One part of the project will continue the PI's study of the relations between Jones-type link polynomials, the topology of essential surfaces in link complements and hyperbolic geometry. Another part, will study the Turaev-Viro three-manifold invariants, their relations to other quantum invariants and the connections of their asymptotics to hyperbolic geometry. A third part will develop methods for recognizing geometric structures on three-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data.  A fourth part will study skein link theory in three-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds.  The project also involves research problems  for graduate students currently working with the PI.