Narrow Results

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

The problem whether the purely cosmetic surgery conjecture holds for any knots is open. Using recent work of Hanselman, the conjecture was proved for any pretzel knots [A. I. Stipsicz and Z. Szabo, Purely cosmetic surgeries and pretzel knots (2020), arXiv:2006.06765]. In this paper, we prove the conjecture for any three strand pretzel knots in a way that does not depend on Heegaard Floer homology.

Let $p$ and $q$ be pairwise coprime with $p\ne 0$ and $q>0$. Let the Brieskorn–Hamm manifold be a homology lens space and the order its the first homology group is $\left|p\right|$. We classify the Brieskorn–Hamm manifold and the resulting $3$-manifold of $p/q$-surgery along a knot $K$ in $S^3$ by using the Lescop invariant.