A well-known theorem of Gordon and Luecke says that knots in the three-sphere are determined by their complements. Boileau asked whether the same is true for null-homotopic knots in three-manifolds. We present two solutions to his conjecture: one found in joint work with Ali Daemi, Tye Lidman, and Nick Rozenblyum, and another found by GPT-6 Astra. Both solutions involve studying SU(2)-character varieties with Yang-Mills gauge theory. The first approach is more conceptual, involving singularity theory and a sheaf quantization approach to the local Floer homology of Lagrangian branes. The sheaf is a real analogue of the vanishing cycle sheaves from complex geometry. As a byproduct of our theory, we construct G-Casson invariants and G-local Floer homology for 3-manifolds, for any reductive Lie group G.