Heegaard Floer correction terms of $(+1)$-surgeries along $(2,q)$-cablings

TL;DR

This paper estimates the Heegaard Floer correction term $d_1$ for (+1)-surgeries along $(2,q)$-cabled knots, revealing that the estimate is independent of the knot type and highlighting the weak relationship between $d_1$ and the $ au$-invariant.

Contribution

It provides a knot-type independent estimate for the correction term $d_1$ for $(2,q)$-cabled knots and explores its relation to the $ au$-invariant.

Findings

The estimate for $d_1$ does not depend on the knot type.

Equality holds for knots in a class containing all negative knots.

The relationship between $d_1$ and the $ au$-invariant is generally weak.

Abstract

The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot concordance invariant $d_1$. In this paper, we estimate $d_1$ for the $(2,q)$-cable of any knot $K$. This estimate does not depend on the knot type of $K$. If $K$ belongs to a certain class which contains all negative knots, then equality holds. As a corollary, we show that the relationship between $d_1$ and the Heegaard Floer $\tau$-invariant is very weak in general.