Petal diagram from simple braids

Abstract

We construct petal diagrams from simple braids. This approach allows us to confirm a conjecture proposed by Kim, No and Yoo, which states that the petal number of the nontrivial torus knot $T_{r,s}$ ($r<s$) is at most $2s-2\lfloor \frac{s}{r}\rfloor +1$. As a consequence, we deduce that the petal number of a nontrivial torus knot $T_{r,s}$ is equal to $2s-1$ if and only if $r<s<2r$.

8 pages, 4 figures