OPEN-CLOSED TQFTS EXTEND KHOVANOV HOMOLOGY FROM LINKS TO TANGLES

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  • AARON D. LAUDA
    Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027–4406, USA
  • HENDRYK PFEIFFER
    Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V2T 1Z2, Canada

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<jats:p> We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A<jats:sup> op </jats:sup> for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant. </jats:p>

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