Knot and 3-manifold invariants and Dehn surgery

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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 relationbetween 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 developedover 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 couldlead to progress on the open question of whether these invariantsdistinguish all knots. She would also like to search for relations between the Jones polynomial of knots and the fundamental group of the 3-manifoldsobtained by Dehn surgery on knots.The research of the project lies in the area of 3-dimensional topology thecentral objects of study of which are spaces called 3-manifolds. A3-manifold is an object that locally looks like the ordinary 3-dimensionalspace but whose global structure can be complicated. A main goal of3-dimensional topology is to understand these structures and achieve aclassification of 3-manifolds. An important part of 3-dimensional topologyis also the study of knots (loops embedded in some tangled way in3-manifolds) and their classification. One of the ways that topologists havebeen approaching these problems is through the use of ``invariants". In therecent years, ideas originated in physics, lead mathematicians to thediscovery of a variety of invariants of knots and 3-manifolds. The centraltheme of the PI's project is to understand the properties of theseinvariants, using ideas from traditional 3-dimensional topology and fromphysics, and investigate the extent to which they distinguishknots and 3-manifolds.