Petal number of torus knots using superbridge indices

Abstract

A petal projection of a knot $K$ is a projection of a knot which consists of a single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number $p(K)$ is the minimum number of loops required to represent the knot $K$ as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})=2s-1$$ when $1<r<s$ and $r\equiv 1\mathrm{mod}s-r$. Furthermore, we also find the upper bound of the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})\le 2s-2\lfloor \frac{s}{r}\rfloor +1$$ when $s\equiv \pm 1\mathrm{mod}r$.