The irreducible tangles are persistent conjecture

A tangle is a diagram of properly embedded arcs in a ball, with the relevant closures obtained by joining its endpoints. A tangle is rationally irreducible if no ambient isotopy together with adjacent end twisting can turn it into a tangle with fewer crossings, and it is neither an infinity tangle nor a zero tangle. An irreducible tangle is rationally irreducible, has no local knots, and has the property that whenever its numerator or denominator closure has one component, that closure is a non-trivial knot.

Irreducible tangles are persistent conjecture. Every irreducible tangle is a persistent tangle.

Here a persistent tangle is one whose appearance as a subtangle of a knot diagram forces the whole diagram to be knotted. The conjecture concerns the prevalence of such tangles and is presented as an open claim in the paper; the supplied text gives no resolution.