Principal Investigator: Efstratia Kalfagianni
The PI will study the ``finite type" invariants of knots and 3-manifolds by using techniques from classical 3-dimensional topology and search for geometric information encoded in these invariants. First, she will continue her work on developing the theory of finite type invariants for knots and links in arbitrary 3-manifolds by using techniques from the theory of atoroidal decompositions of 3-manifolds. She also plans to work on relating the Vassiliev knot invariants to properties of Seifert surfaces spanned by the knots and to intrinsic invariants of the knot complement. This will extend classical results about the topology of the Alexander polynomial to the Jones polynomial and its generalizations. Finally, the PI plans to search for relations between the Jones polynomial of a knot and the fundamental group of 3-manifolds obtained by Dehn surgery on the knot.
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. A main goal of 3-dimensional topology is to understand these structures and achieve a classification of 3-manifolds. 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 traditional 3-dimensional topology and from physics, and to look for applications to the aforementioned classification problems.
