Covering, Correspondence and Noncommutative Geometry

ISBN-13: 9783845412627 / Angielski / Miękka / 108 str.

Ahmad Zainy Al-Yasry

Opis

We construct an additive category whose objects are embedded graphs (or in particular knots) in the 3-sphere and where morphisms are formal linear combinations of 3-manifolds. Our definition of correspondences relies on the Alexander branched covering theorem, which shows that all compact oriented 3-manifolds can be realized as branched coverings of the 3-sphere, with branched locus an embedded (not necessarily connected) graph. The way in which a given 3-manifold is realized as a branched cover is highly not unique. An interesting homology theory for knots and links that we consider here is the one introduced by Khovanov. We recall the basic definition and properties of Khovanov homology and we give some explicit examples of how it is computed for very simple cases such as the Hopf link. We also recall, the construction of the cobordism group for links and for knots and their relation. We then consider the question of constructing a similar cobordism group for embedded graphs in the 3-sphere.

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O autorze

Dr.Ahmad Zainy Al-Yasry is a lecturer in the Department of Mathematics, College of Science for Women, University of Baghdad. His BSc and MSc by University of baghdad. He was awarded the Ph.D. in Mathematics by University of Bonn and Max-Planck Institute for Mathematics. His field of interest include Low Dimensional Topology and AlgebraicTopology