Growth of excess for unknot petal diagrams

Determine whether the excess of unknot petal diagrams is bounded by a function of the number of petals, how rapidly such a function must grow, and whether the maximal excess among diagrams with N petals tends to infinity.

Background

The paper defines the excess of an unknot petal permutation as the minimum number of additional petals required during a sequence of allowed moves that reduces the permutation to the one-petal diagram. Exhaustive computations show that all unknot diagrams with at most 7 petals simplify monotonically, while some 9-petal diagrams have positive excess: two classes have excess 2 and one class has excess 4. The question asks for general bounds and asymptotic behavior of this complexity measure as the number of petals increases.

References

Is the excess of unknot petal diagrams bounded by a function of the number of petals, and how fast must such a function grow? Does the maximal excess over $N$-petal diagrams tend to infinity?

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

How common are hard diagrams? With $9$ petals, $108$ of the $24992$ unknot words (about $0.43\%$) are $M$-hard. What is the proportion for larger $N$, for instance in the Petaluma model ?

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