Surgery on links of linking number zero and the Heegaard Floer $d$-invariant

Abstract

We study Heegaard Floer homology and various related invariants (such as the h -function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the h -function, the Sato–Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer d -invariants of integral surgeries on two-component L-space links of linking number zero in terms of the h -function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the \nu^{+} -invariants of the knots obtained from blowing down the components. We give a proof of a skein inequality for the d -invariants of +1 surgeries along linking number zero links that differ by a crossing change. We also describe bounds on the smooth four-genus of links in terms of the h -function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.