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Abstract

We employ the $sl(2)$ foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of $U_q(sl(2))$. We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is defined over the Gaussian integers $\mathbb{Z}[i]$ (and more generally over $\mathbb{Z}[i][a,h]$, where $a,h$ are formal parameters), and enhances the existing categorifications of the colored Jones polynomial.