## Algebraic & Geometric Topology

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

Carmen Livia Caprau

#### Abstract

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

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

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

https://projecteuclid.org/euclid.agt/1513796842

Digital Object Identifier
doi:10.2140/agt.2008.8.729

Mathematical Reviews number (MathSciNet)
MR2443094

Zentralblatt MATH identifier
1148.57016

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796842