#### Volume 8, issue 2 (2008)

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

### Carmen Livia Caprau

Algebraic & Geometric Topology 8 (2008) 729–756
##### Abstract

We construct a bigraded cohomology theory which depends on one parameter $a$, and whose graded Euler characteristic is the quantum $\mathfrak{s}\mathfrak{l}\left(2\right)$ link invariant. We follow Bar-Natan’s approach to tangles on one side, and Khovanov’s $\mathfrak{s}\mathfrak{l}\left(3\right)$ theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov $\mathfrak{s}\mathfrak{l}\left(2\right)$ 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 $\mathfrak{s}\mathfrak{l}\left(2\right)$ theory.

##### Keywords
categorification, cobordisms, Euler characteristic, Jones polynomial, functoriality, Khovanov homology, knots and links, movie moves, webs and foams
##### Mathematical Subject Classification 2000
Primary: 57M27, 57M25
Secondary: 18G60
##### Publication
Received: 9 January 2008
Accepted: 28 January 2008
Published: 25 May 2008
##### Authors
 Carmen Livia Caprau Department of Mathematics California State University 5245 North Backer Avenue M/S PB 108 Fresno CA 93740-8001 USA http://zimmer.csufresno.edu/~ccaprau