Heegaard Floer homology and knots determined by their complements

TL;DR

This paper explores how Heegaard Floer homology can determine knots by their complements, establishing bounds on surgeries and characterizations of knots in various 3-manifolds.

Contribution

It introduces new bounds on surgeries producing the same manifold and proves knots are determined by their complements in several classes of 3-manifolds.

Findings

Knots in certain homology spheres are determined by their complements.

Null-homologous knots in specific homology $ ext{RP}^3$'s are uniquely identified by their complements.

Knots in some lens spaces and the Brieskorn sphere $ ext{Σ}(2,3,7)$ are also determined by their complements.

Abstract

In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology $\mathbb{R}P^3$'s are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in $L$-spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than $1$ in the Brieskorn sphere $\Sigma(2,3,7)$ are also determined by their complements.