A three-variable transcendental invariant of planar knotoids via Gauss diagrams

TL;DR

This paper introduces a new three-variable transcendental invariant for planar knotoids, extending knot invariants, which helps analyze their properties and Gordian distances.

Contribution

It presents a novel invariant based on Gauss diagrams that is a Vassiliev invariant of order one for planar knotoids.

Findings

The invariant is well-defined and possesses specific properties.

It provides lower bounds on Gordian distances between knotoids.

The invariant extends classical knot invariants to the knotoid setting.

Abstract

As a generalization of the classical knots, knotoids are equivalence classes of immersions of the oriented unit interval in a surface. In recent years, a variety of invariants of spherical and planar knotoids have been constructed as extensions of invariants of classical and virtual knots. In this paper we introduce a three-variable transcendental invariant of planar knotoids which is defined over an index function of a Gauss diagram. We describe properties of this invariant and show that it is a Vassiliev invariant of order one. We also discuss the Gordian distance between planar knotoids and provide lower bounds on the Gordian distance of homotopic planar knotoids by using the transcendental invariant.