Three conditions that turn out to be equivalent
The paper's main theorem concerns finitely generated quandles Q and R and states that three conditions are equivalent. First, for every finite quandle X, the number of homomorphisms from Q to X equals the number from R to X. Second, the profinite completions of Q and R are continuously isomorphic. Third, the profinite completions of the peripheral systems associated to Q and R are equivalent.
This is stronger than a one-directional implication. It says the colouring-count data, the profinite quandle, and the profinite peripheral system all encode the same information about a finitely generated quandle, so none of the three loses anything the others retain within this class.
Colouring counts as a complete invariant of the profinite quandle
A colouring of a link by a finite quandle X is a morphism from the fundamental quandle of the link to X, so the number of colourings is the number of such morphisms. The paper works in the broader setting of topological algebras over a Lawvere theory and generalises a theorem previously known for groups: for a topologically finitely generated algebra X, the numbers of morphisms from X to each finite discrete algebra determine the set of isomorphism classes of finite quotients of X, and that set in turn determines the profinite completion up to continuous isomorphism.
The finiteness ingredient is that a topologically finitely generated algebra has only finitely many open congruences of index n for each integer n at least 1. Without that, the reconstruction of the profinite completion from colouring counts would not go through.
Building the bridge from quandles to peripheral systems
The second half of the proof develops topological and profinite quandle theory, including the enveloping group of a quandle in both settings. The paper constructs the profinite peripheral system associated to a profinite quandle R, tied to the action of the profinite enveloping group of R on R, and extends Joyce's correspondence between quandles and peripheral systems to the profinite world.
The technical condition is that the profinite quandle's set of orbits must be finite and discrete; this holds, for instance, for the profinite completion of a finitely generated discrete quandle. Under that condition, giving such a profinite quandle is equivalent to giving its profinite peripheral system, and for a quandle Q the profinite peripheral system of its profinite completion is the profinite completion of the peripheral system of Q.
Reformulating the completeness conjecture as profinite rigidity
The classical question is whether finite quandle colourings distinguish links: if two links cannot be obtained from one another by taking reversed mirror images of split components, is there a finite quandle whose colouring count differs for the two links? The paper shows this conjecture is equivalent to a profinite rigidity statement: under the same hypothesis on the two links, the profinite completions of their peripheral systems are not equivalent.
The peripheral system of a link records the fundamental group of the complement together with, for each component, a peripheral subgroup and a meridian. Its profinite completion replaces the group by its profinite completion, each meridian by its image, and each peripheral subgroup by the closure of its image. The equivalence of the two conjectures follows by applying the main theorem to fundamental quandles of links, since the number of colourings of a link by a finite quandle X is the number of morphisms from its fundamental quandle to X.
Why the peripheral completion could be the weaker invariant
The profinite completion of the peripheral system is a stronger invariant than the profinite completion of the fundamental group together with the number of components, because it retains the peripheral data. The completion process can still lose information, and the conjecture asserts that in this setting it does not.
The paper notes that profinite rigidity has been conjectured for prime knots and is known for the figure-eight knot and torus knots, and more recently for 2-bridge knots and 3-tangle Montesinos links. It also cites a result that every link is determined by the profinite completion of its fundamental group up to a finite number of possibilities. These are context for how much the peripheral completion is expected to see, not consequences of the main theorem.
A profiniteness criterion using inner automorphisms
For groups, being profinite is a purely topological property, since profinite groups are exactly the compact totally disconnected ones. The paper shows this fails for quandles with an explicit example: a topological quandle that is a profinite space but not residually finite, because its family of inner automorphisms is not equicontinuous.
The replacement criterion is that a topological quandle Q is profinite if and only if it is a profinite space and its group of inner automorphisms is compact. The paper also studies the closure of the inner automorphism group in the homeomorphism group and shows it is a profinite group when Q is profinite. This closure is the object the paper argues is the relevant one for profinite quandle theory, rather than the profinite completion of the inner automorphism group.
The equivalence is stated for finitely generated quandles
The main theorem is stated for finitely generated quandles, and the reconstruction of the profinite completion from colouring counts is proved for topologically finitely generated algebras. The profinite peripheral system construction requires the profinite quandle's orbit set to be finite and discrete, a condition the paper notes is satisfied by the profinite completion of a finitely generated discrete quandle.
The paper also corrects earlier work: for inner automorphisms, the claim that the inner automorphism group of a profinite quandle is profinite is false, even when the quandle is topologically finitely generated, and the paper gives a finitely generated counterexample. The correct statement concerns the closure of the inner automorphism group in the homeomorphism group. Readers should therefore treat the equivalence as a statement about finitely generated quandles and their profinite completions, not as an unrestricted claim about all topological quandles.
Sources used in this answer
Profinite rigidity and quandle colourings
For finitely generated quandles Q and R, equal numbers of colourings by every finite quandle, continuously isomorphic profinite completions, and equivalent profinite completions of the associated peripheral systems are equivalent conditions, reformulating the completeness of finite quandle colourings as a profinite rigidity question.