# Khovanov-Rozansky homology for infinite multi-colored braids

## TL;DR

This paper introduces a limiting version of Khovanov-Rozansky homology for semi-infinite multi-colored braids, extending categorification results to infinite braid structures and broadening the understanding of their algebraic properties.

## Contribution

It defines a new limiting homology for infinite multi-colored braids and demonstrates its role as a categorification of a highest-weight projector, extending prior work on infinite twist braids.

## Key findings

- Limiting homology categorifies highest-weight projectors.
- Extension of categorification to semi-infinite and bi-infinite braids.
- Generalization of previous results on infinite twist braids.

## Abstract

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. 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 multi-colored braids.