Recognizing oriented knots from finite quotients with one marked element

Abstract

For an oriented tame knot $K$, let $G_K$ be its group, with positive meridian $μ_K$ and preferred longitude $λ_K$. We prove that the profinite completion $\widehat{G_K}$, marked by the conjugacy class of $μ_Kλ_K$, determines $K$ up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.