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Covering, Correspondence and Noncommutative Geometry,Used
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We construct an additive category whose objects are embedded graphs (or in particular knots) in the 3sphere and where morphisms are formal linear combinations of 3manifolds. Our definition of correspondences relies on the Alexander branched covering theorem, which shows that all compact oriented 3manifolds can be realized as branched coverings of the 3sphere, with branched locus an embedded (not necessarily connected) graph. The way in which a given 3manifold 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 3sphere.
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