Narrow Results

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsváth and Szabó assume arbitrarily large positive and negative values. As a consequence, we generalize a result by Greene and Watson by proving, for every odd number $\mathrm{\Delta }\ge 5$, the existence of infinitely many non-quasi-alternating homologically thin knots with determinant ${\mathrm{\Delta }}^2$, and a result by Hoffman and Walsh concerning the existence of hyperbolic weight $1$ manifolds, that are not surgery on a knot in $S^3$.