The knot complement problem for null-homotopic knots

TL;DR

This paper proves that in certain 3-manifolds, null-homotopic knots are uniquely determined by their complements, using advanced techniques in Floer homology and representation varieties.

Contribution

It establishes a new result confirming that null-homotopic knots are determined by their complements in specific 3-manifolds, addressing a Kirby Problem.

Findings

Null-homotopic knots are determined by their complements in certain 3-manifolds.

Uses instanton Floer homology and SU(2)-representation varieties.

Answers a Kirby Problem for a special class of 3-manifolds.

Abstract

We prove that for three-manifolds satisfying a certain algebraic condition on their fundamental group, null-homotopic knots are determined by their complements. This answers a Kirby Problem posed by Boileau for this special case of 3-manifolds. The argument uses techniques in instanton Floer homology and SU(2)-representation varieties.