Khovanov-Rozansky $\mathfrak{sl}_N$-homology for periodic links

TL;DR

This paper demonstrates that the Khovanov-Rozansky $ ext{sl}_N$-homology for periodic links admits a natural $ ext{Z}_m$ group action, providing new tools for analyzing link periodicity.

Contribution

It introduces a $ ext{Z}_m$-action on $ ext{sl}_N$-homology for periodic links and applies this to establish a periodicity criterion analogous to Borodzik--Politarczyk's.

Findings

$ ext{sl}_N$-homology admits a $ ext{Z}_m$-action for $m$-periodic links

Established an $ ext{sl}_N$-homology-based periodicity criterion

Extended periodicity analysis beyond Khovanov homology

Abstract

For an $m$-periodic link $L$, we show that the Khovanov-Rozansky $\mathfrak{sl}_N$-homology carries an action of the group $\mathbb{Z}_m$. As an example of applications, we prove an analog of the periodicity criterion of Borodzik--Politarczyk using $\mathfrak{sl}_N$-homology instead of Khovanov homology.