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Open-closed TQFTs extend Khovanov homology from links to tangles. (English) Zbl 1161.57018

This paper puts forward a new extension of Khovanov homology from links to tangles, using a kind of extended 2-dimensional TQFT called open-closed TQFT. This extension of Khovanov homology is physically well-motivated – it can be understood as coming from “correlators” that the open-closed TQFT assigns to a collection of D-branes determined by tangles, analogously to the way Khovanov homology for links can be understood as coming from “correlators” that a certain closed-string TFT assigns to a collection of worldsheets determined by links.
An open-closed TQFT is a symmetric monoidal functor from Bar-Natan’s “picture world” [D. Bar-Natan, Geom. Topol. 9, 1443-1499 (2005; Zbl 1084.57011)] to a custom-made algebraic structure called a knowledgeable Frobenius algebra. As a result of the Cardy condition, one needs to work over a field of finite characteristic. If the commutative part of the knowledgeable Frobenius algebra is the centre of its symmetric part, the open-closed TQFT comes from a state-sum model, and we have an algebraic operation corresponding to composition of tangles. Examples are given of state-sum open-closed TQFT’s which determine characteristic \(p\) Khovanov homology for links and Rasmussen’s \(s\)–invariant.

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

Citations:

Zbl 1084.57011
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References:

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