Petal grid diagrams of torus knots

Abstract

A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number [Formula: see text] of a knot [Formula: see text] is the minimum number of loops among all petal diagrams of [Formula: see text]. Let [Formula: see text] denote the [Formula: see text]-torus knot for relatively prime integers [Formula: see text]. Recently, Kim et al. proved that [Formula: see text] whenever [Formula: see text]. They conjectured that the inequality holds without the assumption [Formula: see text]. They also showed that [Formula: see text] whenever [Formula: see text] and [Formula: see text]. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that [Formula: see text] holds for any [Formula: see text]. We also show that [Formula: see text] holds for any [Formula: see text]. Our proofs construct petal grid diagrams for any torus knots.