Petal number of torus knots using superbridge indices

Abstract

A petal projection of a knot [Formula: see text] 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 [Formula: see text] is the minimum number of loops required to represent the knot [Formula: see text] 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 [Formula: see text] as follows: [Formula: see text] when [Formula: see text] and [Formula: see text] mod [Formula: see text]. Furthermore, we also find the upper bound of the petal number of [Formula: see text] as follows: [Formula: see text] when [Formula: see text] mod [Formula: see text].