HOMFLYPT Homology over $\mathbb{Z}_2$ Detects Unlinks

TL;DR

This paper proves that the HOMFLYPT homology over  can detect unlinks by leveraging spectral sequences and existing results on Khovanov homology, advancing the understanding of link invariants in knot theory.

Contribution

It demonstrates that the -graded structure of HOMFLYPT homology detects unlinks, extending previous detection results from Khovanov homology.

Findings

HOMFLYPT homology over  detects unlinks

Spectral sequence techniques are effective in link detection

Extension of Khovanov homology detection results

Abstract

We apply the Rasmussen spectral sequence to prove that the $\mathbb{Z}^3$-graded vector space structure of the HOMFLYPT homology over $\mathbb{Z}_2$ detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the $\mathbb{Z}^2$-graded vector space structure of the Khovanov homology over $\mathbb{Z}_2$ detects unlinks.