Vassiliev invariants for virtual knotoids

TL;DR

This paper introduces a new 0-smoothing invariant for virtual knotoids, proves it is a Vassiliev invariant of order one, and compares its informational content to existing invariants, showing it provides less information.

Contribution

The paper defines a novel 0-smoothing invariant for virtual knotoids and establishes its properties as a Vassiliev invariant of order one, extending Turaev's singular virtual string invariant.

Findings

The 0-smoothing invariant is a Vassiliev invariant of order one.

It provides less information than the gluing invariant for virtual knotoids.

The invariant is constructed using local modifications at classical crossings.

Abstract

In this paper, we introduce the 0-smoothing invariant $\mathcal{F}$ of virtual knotoids constructed from local modification at classical crossings, which take values in a free $\mathbb Z$-module generated by non-oriented flat virtual knotoids. We prove that $\mathcal{F}$ is a Vassiliev invariant of order one. It was observed by Henrich that smoothing invariant she constructed for virtual knots provides less information than the gluing invariant. We demonstrate the same property for the 0-smoothing invariant of virtual knotoids: $\mathcal{F}$ provides less information than the gluing invariant introduced by Petit. To prove this result, we use the extension of the singular based matrix invariant originally introduced by Turaev for singular virtual strings.