Colored Khovanov–Rozansky homology for infinite braids

Abstract

We show that the limiting unicolored $\mathfrak{s}\mathfrak{l}\left(N\right)$ Khovanov–Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored homfly–pt Khovanov–Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the homfly–pt homology of any braid positive link and the stable homfly–pt homology of the infinite torus knot as computed by Hogancamp.