Petal grid diagrams of torus knots

A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number $p(K)$ of a knot $K$ is the minimum number of loops among all petal diagrams of $K$. Let $T_{n,s}$ denote the $(n,s)$-torus knot for relatively prime integers $2\le n<s$. Recently, Kim, No and Yoo proved that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn\right\rfloor+1$ whenever $s\equiv \pm 1\bmod n$. They conjectured that the inequality holds without the assumption $s\equiv \pm 1\bmod n$. They also showed that $p(T_{n,s})=2s-1$ whenever $2\le n<s<2n$ and $n\equiv 1\bmod s-n$. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn\right\rfloor+1$ holds for any $2\le n<s$. We also show that $p(T_{n,s})=2s-1$ holds for any $2\le n<s<2n$. Our proofs construct petal grid diagrams for any torus knots.