Correction terms of double branched covers and symmetries of immersed curves

TL;DR

This paper uses immersed curves in bordered Floer homology to analyze d-invariants of double branched covers of arborescent links, introducing a new invariant and relating it to known invariants and symmetries.

Contribution

It introduces a new invariant elta_{sym} for bordered bZ_2-homology solid tori and relates it to d-invariants and arta-invariants, linking symmetries of links to 3-manifold invariants.

Findings

d-invariants of double branched covers are determined by link signatures under certain conditions

elta_{sym} invariant relates to d-invariants and arta-invariants

Symmetries of links influence the d-invariants of their double branched covers

Abstract

We use the immersed curves description of bordered Floer homology to study $d$-invariants of double branched covers $\Sigma_2(L)$ of arborescent links $L \subset S^3$. We define a new invariant $\Delta_{sym}$ of bordered $\mathbb{Z}_2$-homology solid tori from an involution of the associated immersed curves and relate it to both the $d$-invariants and the Neumann-Siebenmann $\bar\mu$-invariants of certain fillings. We deduce that if $L$ is a 2-component arborescent link and $\Sigma_2(L)$ is an L-space, then the spin $d$-invariants of $\Sigma_2(L)$ are determined by the signatures of $L$. By a separate argument, we show that the same relationship holds when $L$ is a 2-component link that admits a certain symmetry.