On Link Homology Theories from Extended Cobordisms

Abstract : This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel-Smith construction, and the odd Khovanov homology fit into this setting. Moreover, we prove that any EQFT based on a Z_2-extension of the embedded cobordism category which coincides with Khovanov after reducing the coefficients modulo 2, gives rise to a link invariant homology theory isomorphic to those of Khovanov.
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https://hal.archives-ouvertes.fr/hal-00426628
Contributor : Emmanuel Wagner <>
Submitted on : Tuesday, October 27, 2009 - 10:07:16 AM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

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  • HAL Id : hal-00426628, version 1
  • ARXIV : 0910.5050

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Anna Beliakova, Emmanuel Wagner. On Link Homology Theories from Extended Cobordisms. 2009. ⟨hal-00426628⟩

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