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

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.