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Quantum Topology

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Volume 1, Issue 4, 2010, pp. 379–398
DOI: 10.4171/QT/9

Published online: 2010-10-17

On link homology theories from extended cobordisms

Anna Beliakova[1] and Emmanuel Wagner[2]

(1) University of Zurich
(2) Institut de Mathématiques de Bourgogne, Dijon, France

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 ℝ3. 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 (strict) EQFT based on a ℤ2-extension of the embedded cobordism category that coincides with Khovanov homology after reducing the coefficients modulo 2 gives rise to a link invariant homology theory isomorphic to that of Khovanov.

Keywords: 2-cobordism, Jones polynomial, cohomology of categories

Beliakova Anna, Wagner Emmanuel: On link homology theories from extended cobordisms. Quantum Topol. 1 (2010), 379-398. doi: 10.4171/QT/9