  In this paper, we study the skein exact sequence for links via the exact
surgery triangle of link Floer homology and compare it with other skein exact
sequences given by Ozsv\'ath and Szab\'o. As an application, we use the skein
exact sequence to study the splitting number and splitting maps for links. In
particular, we associate the splitting maps for the torus link $T(n, n)$ to
integer points in the $(n-1)$-dimensional permutahedron, and obtain the link
Floer homology of an $n$-component homology nontrivial unlink in $S^{1}\times
S^{2}$.
