Split link detection for sl ( P ) $\mathfrak {sl}(P)$ link homology in characteristic P $P$

We provide a sufficient condition for splitness of a link in terms of its reduced $\mathrm{sl}\left(N\right)$ link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If $N$ is prime and the coefficient field is of characteristic $N$, then the sufficient condition for splitness is also necessary. When $N=2$, we recover Lipshitz–Sarkar's split link detection result for Khovanov homology with $Z/2$ coefficients.