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Khovanov’s homology for tangles and cobordisms. (English) Zbl 1084.57011

This is an expository paper on Khovanov’s homology theory with an essential change of approach which allows a substantial generalization with respect to tangles. Khovanov’s construction starts with a formal complex of smoothings of link diagrams and of cobordisms between them. In the second step a functor \(F\) is constructed (a TQFT) into an orthodox complex of modules and homomorphisms yielding a homology which is proved to be a link invariant and whose Euler characteristic is the Jones polynomial. The new approach proves invariance at a stage before applying the functor \(F\). The proof rests on local arguments which allows a generalized composition of tangles.
The paper is well readable and suitable for an introduction to the subject. It is moreover furnished with a wealth of enlightning pictures.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
55N99 Homology and cohomology theories in algebraic topology
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References:

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