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Quantum Topology

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Volume 11, Issue 2, 2020, pp. 323–378
DOI: 10.4171/QT/137

Published online: 2020-06-21

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

Eugene Gorsky[1], Beibei Liu[2] and Allison H. Moore[3]

(1) University of California, Davis, USA and National Research University Higher School of Economics, Moscow, Russia
(2) Max-Planck-Institut für Mathematik, Bonn, Germany
(3) Virginia Commonwealth University, Richmond, USA

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.

Keywords: Link surgery, Heegaard Floer homology, $d$-invariant, $h$-function, concordance, four-genus

Gorsky Eugene, Liu Beibei, Moore Allison: Surgery on links of linking number zero and the Heegaard Floer $d$-invariant. Quantum Topol. 11 (2020), 323-378. doi: 10.4171/QT/137