Murasugi sum and extremal knot Floer homology

The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $\tau=g$ for each summand if and only if $\tau=g$ holds for the Murasugi sum (with $\tau$ and $g$ defined appropriately for multi-component links). Some applications are presented.