Algebraic & Geometric Topology

sl(2) tangle homology with a parameter and singular cobordisms

Carmen Livia Caprau

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We construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan’s approach to tangles on one side, and Khovanov’s sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl(2) invariant (or Lee’s modification of it) corresponds to a=0 (or a=1). In particular, the construction naturally resolves the sign ambiguity in the functoriality of Khovanov’s sl(2) theory.

Article information

Algebr. Geom. Topol., Volume 8, Number 2 (2008), 729-756.

Received: 9 January 2008
Accepted: 28 January 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

categorification cobordisms Euler characteristic Jones polynomial functoriality Khovanov homology knots and links movie moves webs and foams


Caprau, Carmen Livia. sl(2) tangle homology with a parameter and singular cobordisms. Algebr. Geom. Topol. 8 (2008), no. 2, 729--756. doi:10.2140/agt.2008.8.729.

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