Knot and 3-manifold invariants and Dehn surgery

Michigan State University, East Lansing MI

Investigators

Abstract

The PI proposes to explore the applicability of classical 3-dimensional topology techniques in the study of the ``finite type" invariants of knots and 3-manifolds. She would like to investigate the relation between intrinsic invariants (e.g. knot genus, satellite structures) of knots that cannot be distinguished by their finite type invariants. For this she proposes to use the general machinery that has been developed over the years to understand how the topology of 3-manifolds changes under surgery. The resulting work will shed some light on how the finite type knot invariants relate to the topology of the knot complement and could lead to progress on the open question of whether these invariants distinguish all knots. She would also like to search for relations between the Jones polynomial of knots and the fundamental group of the 3-manifolds obtained by Dehn surgery on knots. 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 investigate the extent to which they distinguish knots and 3-manifolds.