Murasugi sum and extremal knot Floer homology

TL;DR

This paper investigates how extremal knot Floer homology behaves under Murasugi sum, establishing a graded isomorphism and conditions relating tau and g invariants, with applications to link theory.

Contribution

It introduces a graded isomorphism for extremal knot Floer homology under Murasugi sum and characterizes tau and g invariants for summed links.

Findings

Established a graded version of Ni's isomorphism for Murasugi sums.

Proved tau=g condition for summands iff it holds for the sum.

Presented applications in link theory.

Abstract

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.