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.

💡 Research Summary

The paper introduces a new order‑one Vassiliev invariant for virtual knotoids, denoted by 𝔽, constructed by applying a 0‑smoothing operation at each classical crossing of a virtual knotoid diagram. The authors first recall the necessary background on knotoids, virtual knotoids, flat knotoids, and the affine index polynomial, setting the stage for their construction.

A virtual knotoid diagram consists of classical and virtual crossings on an oriented arc with distinct tail and head. By ignoring the over/under information at a crossing (i.e., replacing it with a flat crossing) one obtains a flat virtual knotoid. The 0‑smoothing at a classical crossing c replaces the crossing by a pair of arcs that bypass each other, producing two flat virtual knotoids K₊(c) and K₋(c) depending on the smoothing direction. The invariant 𝔽 is defined as the ℤ‑linear combination

 𝔽(K) = Σ_{c∈C(K)} sgn(c)·(K₊(c) – K₋(c)),

where C(K) is the set of classical crossings and sgn(c)∈{±1} is the usual crossing sign. The target space of 𝔽 is the free ℤ‑module generated by isotopy classes of non‑oriented flat virtual knotoids.

The authors prove three main properties. First (Theorem 3.1) they show that 𝔽 is invariant under the full set of generalized Reidemeister moves for virtual knotoids (classical, virtual, mixed, and the tail/head slide move Ωᵥ). The proof proceeds by checking each move locally and observing that the contributions of the smoothed diagrams cancel appropriately. Second (Theorem 3.4) they establish that 𝔽 is a Vassiliev invariant of order one. By introducing a singular crossing (a “double point”) and considering the two resolutions (positive and negative), they demonstrate that the alternating sum of 𝔽 over the two resolutions vanishes, which is precisely the defining condition for a degree‑one finite‑type invariant. Third (Theorem 3.3) they analyze the behavior of 𝔽 under orientation reversal and mirroring, showing that 𝔽( K̅ ) = –𝔽(K) and 𝔽( K* ) = 𝔽(K) up to a sign depending on the number of crossings, confirming the expected symmetry properties.

Having established 𝔽 as a legitimate order‑one invariant, the paper turns to a comparison with the “gluing invariant” 𝔾 introduced by Petit, which is known to be the universal order‑one Vassiliev invariant for virtual knots and, by extension, for virtual knotoids. Petit’s construction replaces each classical crossing by a pair of glued arcs, producing a singular virtual knotoid; the resulting combinatorial data are encoded in a singular based matrix (SBM), originally defined by Turaev for singular virtual strings. The SBM records both the sign of each crossing and the way the glued arcs interact, thus retaining more information than the flat smoothing approach.

The central comparison result (Theorem 4.2) shows that the 0‑smoothing invariant 𝔽 carries strictly less information than 𝔾. Concretely, the authors construct a homomorphism from the SBM‑based invariant 𝔾 onto the free ℤ‑module of flat knotoids that sends 𝔾 to 𝔽, proving that any two virtual knotoids distinguished by 𝔽 are also distinguished by 𝔾, but the converse fails. The proof relies on extending Turaev’s singular based matrix theory to the setting of virtual knotoids, carefully adapting the matrix operations to respect the open‑ended nature of knotoids.

The paper concludes by emphasizing the significance of having both a computationally simple invariant (𝔽) and a universal invariant (𝔾) for virtual knotoids. While 𝔽 is easy to compute—requiring only local smoothings and a tally in a free module—it cannot detect certain subtle differences that 𝔾 captures via its matrix data. The authors suggest that future work could explore higher‑order Vassiliev invariants for virtual knotoids, perhaps by considering multiple singularities and extending the SBM framework accordingly, or by combining the strengths of 𝔽 and 𝔾 into a hybrid invariant that balances computational tractability with discriminating power.