# A Reidemeister type theorem for petal diagrams of knots

## TL;DR

This paper introduces a Reidemeister type theorem for petal diagrams of knots, showing that isotopic knots can be related through specific permutation moves, thus providing a combinatorial framework for knot equivalence.

## Contribution

It establishes a set of moves on permutations representing petal diagrams that preserve knot isotopy, proving their sufficiency for relating all diagrams of the same knot.

## Key findings

- Any two permutations representing isotopic knots can be connected by a sequence of defined moves.
- The moves include trivial petal additions and crossing exchanges, which do not alter the knot type.
- The theorem provides a combinatorial approach to classifying knots via petal diagrams.

## Abstract

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do not change the isotopy class of the underlying knot. We prove that any two permutations which represent isotopic knots can be related by a sequence of these moves and their inverses.