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  Open-closed TQFTs extend Khovanov homology from links to tangles

Lauda, A. D., & Pfeiffer, H. (2009). Open-closed TQFTs extend Khovanov homology from links to tangles. Journal of Knot Theory and its Ramifications, 18(1), 87-150. doi:10.1142/S0218216509006793.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-455E-8 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-455F-6
Genre: Journal Article

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Lauda, Aaron D., Author
Pfeiffer, Hendryk1, Author
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1Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24014              

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 Abstract: We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex are modules of a suitable algebra A such that there is one action of A or A^op 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. In all cases in which the algebra A is strongly separable, i.e. for Bar-Natan's theory in any characteristic and for Lee's theory in characteristic other than 2, we also provide the required algebraic operation for the composition of oriented tangles. Just as Khovanov's theory for links can be recovered from Lee's or Bar-Natan's by a suitable spectral sequence, we provide a spectral sequence in order to compute our tangle extension of Khovanov's theory from that of Bar-Natan's or Lee's theory. Thus, we provide a tangle homology theory that is locally computable and still strong enough to recover characteristic p Khovanov homology for links.

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 Dates: 2009-01
 Publication Status: Published in print
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 Identifiers: eDoc: 264874
URI: http://arxiv.org/abs/math/0606331
Other: arXiv:math/0606331v1 [math.GT]
DOI: 10.1142/S0218216509006793
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Title: Journal of Knot Theory and its Ramifications
Source Genre: Journal
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Pages: - Volume / Issue: 18 (1) Sequence Number: - Start / End Page: 87 - 150 Identifier: ISSN: 0218-2165