Vassiliev invariants for virtual knotoids

TL;DR

The paper develops a finite-type framework for virtual knotoids by introducing two smoothing invariants $\mathcal{F}$ and $\mathcal{L}$ and a universal order-one invariant $\mathcal{G}$. It integrates singular virtual knotoids and singular based matrices to prove universality of $\mathcal{G}$ and shows that $\mathcal{G}$ strictly captures information beyond $\mathcal{F}$ and $\mathcal{L}$. The approach combines 0-/1-smoothing, gluing, and SBM techniques to distinguish knotoid classes that smoothing alone cannot differentiate. This yields a robust, algebraically grounded toolkit for analyzing virtual knotoids and their finite-type invariants, with potential implications for flat and singular knot theories.

Abstract

In this paper, we introduce three invariants of virtual knotoids. Two of them are smoothing invariants $\mathcal{F}$ and $\mathcal{L}$ which take values in a free $\mathbb Z$-module generated by non-oriented flat virtual knotoids. We prove that $\mathcal{F}$ and $\mathcal{L}$ are both Vassiliev invariants of order one. Moreover we construct a universal Vassiliev invariant $\mathcal{G}$ of order one of virtual knotoids. We demonstrate that $\mathcal{G}$ is stronger than $\mathcal{F}$ and $\mathcal{L}$. To prove this result, we extend the singular based matrix invariant for singular virtual strings introduced by Turaev and Henrich to singular virtual open strings corresponding to singular flat virtual knotoids with one singular crossing.

Figures (40)

• Figure 1: Examples of knotoid diagrams.
• Figure 2: Reidemeister moves $\Omega_1$, $\Omega_2$ and $\Omega_3$.
• Figure 3: Forbidden knotoid moves $\Phi_{+}$ and $\Phi_{-}$.
• Figure 4: Crossings: (a) classical, (b) virtual, (c) flat and (d) singular.
• Figure 5: Virtual Reidemeister moves $\Omega_1^v$, $\Omega_2^v$, $\Omega_3^v$, the mixed move $\Omega_3^{m}$ and $\Omega_{v}$

Theorems & Definitions (61)

• Definition 2.1
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• Proposition 2.1: GK
• Definition 2.9