Knot and 3-Manifold Invariants, Seifert Surfaces and Dehn Surgery

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AbstractAward: DMS-0104000Principal Investigator: Efstratia KalfagianniThe PI will study the ``finite type" invariants of knots and3-manifolds by using techniques from classical 3-dimensionaltopology and search for geometric information encoded in theseinvariants. First, she will continue her work on developing thetheory of finite type invariants for knots and links in arbitrary3-manifolds by using techniques from the theory of atoroidaldecompositions of 3-manifolds. She also plans to work onrelating the Vassiliev knot invariants to properties of Seifertsurfaces spanned by the knots and to intrinsic invariants of theknot complement. This will extend classical results about thetopology of the Alexander polynomial to the Jones polynomial andits generalizations. Finally, the PI plans to search forrelations between the Jones polynomial of a knot and thefundamental group of 3-manifolds obtained by Dehn surgery on theknot.The research of the project lies in the area of 3-dimensionaltopology the central objects of study of which are spaces called3-manifolds. A 3-manifold is an object that locally looks likethe ordinary 3-dimensional space but whose global structure canbe complicated. A main goal of 3-dimensional topology is tounderstand these structures and achieve a classification of3-manifolds. An important part of 3-dimensional topology is alsothe study of knots (loops embedded in some tangled way in3-manifolds) and their classification. One of the ways thattopologists have been approaching these problems is through theuse of ``invariants". In the recent years, ideas originated inphysics, lead mathematicians to the discovery of a variety ofinvariants of knots and 3-manifolds. The central theme of thePI's project is to understand the properties of these invariants,using ideas from traditional 3-dimensional topology and fromphysics, and to look for applications to the aforementionedclassification problems.