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We define a limiting
${\mathfrak {sl}_N}$
Khovanov–Rozansky homology for semi-infinite positive multicolored braids. For a large class of such braids, we show that this limiting homology categorifies a highest-weight projector in the tensor product of fundamental representations determined by the coloring of the braid. This effectively completes the extension of Cautis’ similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multicolored braids.
We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the 
$(n,n)$-torus links, for which we give an exact answer for all 
$n$. In several cases, our computations verify conjectures of Gorsky et al. relating homology of torus links with Hilbert schemes.
