Profinite peripheral rigidity of links

Establish that if two links cannot be obtained from one another by taking reversed mirror images of split components, then the profinite completions of their peripheral systems are not equivalent.

Background

The profinite completion of a link peripheral system records the profinite completion of the link group together with the closures of the peripheral subgroups and the meridian elements. Passing to profinite completions can potentially lose information, so the conjecture asks whether this completed peripheral data still distinguishes links up to reversed mirror images of split components.

The paper proves that this formulation is equivalent to the finite-quandle colouring conjecture, rather than resolving the conjecture itself.

References

We show that it is equivalent to: \begin{conj} If two links cannot be obtained from one another by taking reversed mirror images of split components, then the profinite completions of their peripheral systems are not equivalent. \end{conj}

— Profinite rigidity and quandle colourings  (2610.06485 - Darn{é}, 5 Oct 2026) in Introduction, paragraph “Main results”