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

TL;DR

This paper investigates Heegaard Floer homology and related invariants for two-component L-space links with linking number zero, providing explicit formulas, characterizations, and bounds relevant to surgeries and four-genus calculations.

Contribution

It introduces explicit relationships between invariants for such links, generalizes formulas for $d$-invariants, and offers new bounds on the smooth four-genus.

Findings

Explicit formulas for $d$-invariants of surgeries on these links.

Characterization of L-space surgery slopes via $ u^{+}$-invariants.

Bounds on the smooth four-genus based on the $h$-function.

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.