Petal number of torus knots using superbridge indices

A petal projection of a knot K is a projection of a knot which consists of 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 Tr,s as follows: p(Tr,s) = 2s - 1 when 1 < r < s and r 1 mod s - r. Furthermore, we also find the upper bound of the petal number of Tr,s as follows: p(Tr,s) ≤ 2s - 2s r + 1 when s ±1 mod r.