Hard petal diagrams for nontrivial knots

Determine whether every nontrivial knot K has petal diagrams that cannot be reduced to a diagram with p(K) petals without first increasing the number of petals.

Background

The paper studies hardness for petal diagrams of the unknot and finds examples that require temporarily increasing the number of petals before they can be simplified. The petal number p(K) is the minimum number of petals in any petal diagram of a knot K. The unresolved question is whether an analogous phenomenon occurs for nontrivial knots: namely, whether some diagrams representing a fixed knot necessarily pass through diagrams of greater petal number before reaching minimal petal number.

References

For a non-trivial knot $K$, are there petal diagrams of $K$ that cannot be reduced to a diagram with $p(K)$ petals without first increasing the number of petals? Dynnikov's theorem has no analogue of this kind for arbitrary knots in the rectangular setting either, so the question is whether petal diagrams behave even worse.

— Some petal diagrams of the unknot are hard  (2610.01596 - Vernitski, 1 Oct 2026) in Section Questions