Volume 19, issue 5 (2019)

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Colored Khovanov–Rozansky homology for infinite braids

Michael Abel and Michael Willis

Algebraic & Geometric Topology 19 (2019) 2401–2438
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 homflypt Khovanov–Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the homflypt homology of any braid positive link and the stable homflypt homology of the infinite torus knot as computed by Hogancamp.

Keywords
Khovanov homology, Khovanov–Rozansky homology, link homology, colored link homology, colored Khovanov–Rozanksy homology, infinite braids, infinite twist
Primary: 57M27