Invariant to certify existence of ribbon or symmetric-union strongly positive amphicheiral knots without doubly symmetric diagrams

Develop a knot invariant that enables a proof of existence of a strongly positive amphicheiral knot that is ribbon—or, more specifically, a symmetric union—that does not admit any doubly symmetric diagram. The invariant should distinguish the property of being representable by a doubly symmetric diagram from being a ribbon knot or a symmetric union within the class of strongly positive amphicheiral knots, thereby certifying at least one explicit counterexample to the implication that ribbon or symmetric-union strongly positive amphicheiral knots must have doubly symmetric diagrams.

Background

The paper systematically enumerates templates for doubly symmetric diagrams and determines all prime knots with doubly symmetric crossing number c_ds up to 18, using diagram moves and filter rules to manage complexity. While the enumeration is successful for up to three self-intersections (n ≤ 3), it becomes impractical for n = 4 due to the large number of cases.

The authors note that even a complete generation of doubly symmetric diagrams up to a certain crossing number would not resolve whether the remaining ‘almost doubly symmetric’ ribbon knots up to 16 crossings are truly doubly symmetric. This motivates an open problem, inherited from the first part of the project, calling for a knot invariant that can separate the property of having a doubly symmetric diagram from being ribbon or a symmetric union within the class of strongly positive amphicheiral knots.