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 $S^3$.

This paper contains two remarks about the application of the $d$-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math.192(3) (2013) 717–750] concerning Conway mutation of alternating links.

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth–Szabó correction terms, as smooth double sliceness obstructions, vanish for both knots. Recently, Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. As an application, we give a new example of such a knot that is distinct from Meier’s knots modulo doubly slice knots.