Petal number of torus knots using superbridge indices

TL;DR

This paper establishes a relationship between superbridge indices and petal numbers of knots, specifically calculating petal numbers for certain torus knots and providing upper bounds for others, advancing knot theory understanding.

Contribution

It introduces a new relation between superbridge index and petal number, and computes petal numbers for specific torus knots, offering bounds for others.

Findings

Petal number of $T_{r,s}$ when $1<r<s$ and $r mod (s-r)=1$ is $2s-1

Upper bound for petal number of $T_{r,s}$ when $s mod r= ext{±}1$ is $2s - 2loor{s/r} + 1$

Established a link between superbridge index and petal number for arbitrary knots.

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 $r \equiv 1 \mod s-r$. Furthermore, we also find the upper bound of the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})\leq2s- 2\Big\lfloor \frac{s}{r} \Big\rfloor +1$$ when $s \equiv \pm 1 \mod r$.