Colored Khovanov-Rozansky homology for infinite braids
Michael Abel, Michael Willis
TL;DR
This paper demonstrates that the limiting unicolored (N) Khovanov-Rozansky chain complex of infinite positive braids categorifies a highest-weight projector, extending previous work and applying to colored HOMFLY-PT homology.
Contribution
It introduces a new categorification of highest-weight projectors for infinite braids and extends the framework to colored HOMFLY-PT homology, connecting finite braid invariants to infinite limits.
Findings
Limiting complexes categorify highest-weight projectors.
Results hold for colored HOMFLY-PT Khovanov-Rozansky homology.
Partial isomorphism between braid positive link homology and stable HOMFLY-PT homology.
Abstract
We show that the limiting unicolored 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.
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