# Parity in Knotoids

## TL;DR

This paper explores the parity concept in knotoids, demonstrating the non-surjectivity of the virtual closure map, introducing a planar parity bracket polynomial, and proving a conjecture about minimal diagrams of knot-type knotoids.

## Contribution

It introduces a planar parity bracket polynomial for knotoids and proves a conjecture on minimal diagrams, advancing the understanding of knotoid invariants and their relation to virtual knots.

## Key findings

- Virtual closure map is not surjective.
- Introduces planar parity bracket polynomial for knotoids.
- Proves minimal diagrams of knot-type knotoids have zero height.

## Abstract

This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.