The Khovanov homology of infinite braids

TL;DR

This paper demonstrates that the limiting Khovanov chain complex of infinite positive braids categorifies the Jones-Wenzl projector, extending Rozansky's work and also applies to related homotopy types.

Contribution

It introduces a new categorification of the Jones-Wenzl projector via infinite braids, extending previous results to more general cases.

Findings

Limiting Khovanov complexes categorify Jones-Wenzl projectors

Results extend Rozansky's categorification to infinite braids

Applicable to Lipshitz-Sarkar-Khovanov homotopy types

Abstract

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.