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

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.