A note on applications of the d-invariant and Donaldson's theorem

TL;DR

This paper discusses the use of the d-invariant and Donaldson's theorem in knot theory, showing their equivalence as obstructions and proposing a replacement of Heegaard Floer homology with Donaldson's theorem in certain proofs.

Contribution

It establishes the equivalence of two obstructions to sliceness and suggests replacing Heegaard Floer homology with Donaldson's theorem in specific knot theory proofs.

Findings

Equivalence of two sliceness obstructions for 2-bridge knots

Proposal to replace Heegaard Floer homology with Donaldson's theorem

Application of Donaldson's theorem to Conway mutation of alternating links

Abstract

This note 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 Invent. Math. 192 (2013) no.3 717-750 concerning Conway mutation of alternating links.