On $d$-invariants and generalised Kanenobu knots

Abstract

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsv\'ath and Szab\'o are arbitrarily large and small. As a consequence, we generalise a result by Greene and Watson by proving, for every odd number $\Delta \geq 5$, the existence of infinitely many non-quasi-alternating homologically thin knots with determinant $\Delta^2$, and a result by Hoffman and Walsh concerning the existence of hyperbolic weight $1$ manifolds that are not surgery on a knot in $S^3$.