New Invariants of Knotoids

TL;DR

This paper introduces new invariants for knotoids, such as the odd writhe and polynomial invariants, and explores their properties and bounds, extending the theory of virtual knotoids.

Contribution

It constructs novel invariants for classical knotoids, including the odd writhe, parity bracket, affine index, and arrow polynomial, and relates them to virtual knotoid theory.

Findings

Affine index polynomial bounds the height of knotoids.

Classical knotoids have symmetric affine index polynomials.

New invariants distinguish knotoids from classical knots.

Abstract

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for classical knotoids in analogy to corresponding invariants of virtual knots. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a classical knotoid. We show that classical knotoids have symmetric affine index polynomials.