Vassiliev invariants for virtual knotoids

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.