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 ν + -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.

Triple linking numbers and Heegaard Floer homology

“…By Theorem 1.6, it thus suffices to know that the link Floer complex determines the Sato-Levine invariant (for links with two components) or Milnor triple linking number (for links with three components). This is shown in [GLM20] for links with two components and Theorem 1.2 for links with three components.…”

“…Example 3.7. For two-component algebraically split links the invariant a 2 pLq agrees with the Sato-Levine invariant βpLq up to sign [GLM20,Stu84]. Equation (10) now gives an explicit formula for βpLq in terms of the link Floer complex for L. Moreover, if L is a two-component algebraically split L-space link with unknotted components, then βpLq " 0 implies L is the unlink (see [GLM20, Corollary 6.4]).…”

“…We review the definition of the h-function for oriented links L Ă S 3 , as defined by the first author and Némethi [GN15]. We will quote without proof several technical lemmas regarding its properties; proofs of these statements can be found in either [BG18], [GLM20], or both.…”

“…In previous work of the first, third, and fourth authors [GLM20], it is shown that Heegaard Floer homology is able to see the Sato-Levine invariant β of an algebraically split two-component link. In this paper, we address Problem 1.1 to study several appearances of the Milnor triple linking number µ 123 [Mil57] in the Heegaard Floer theory of links and three-manifolds.…”

Fibered and strongly quasi-positive $L$-space links

“…We refer to [2,11] for general formulas. The explicit formula for links with one and two components can be found in [10].…”