Two-Variable Parity Polynomial for Virtual Knotoids

Two-Variable Parity Polynomial for Virtual Knotoids Full article

Journal

Mathematics
, E-ISSN: 2227-7390

Output data

Year: 2026, Volume: 14, Number: 14, Article number : 2536, Pages count : 26 DOI: 10.3390/math14142536

Tags

virtual knotoids; parity of crossings; Vassiliev invariant of order one

Authors

Ding Siqi ¹ , Gao Suo ² , Lei Fengchun ^3,1 , Li Fengling ¹ , Vesnin Andrei ^4,5

Affiliations

1	School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2	School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China
3	Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
4	Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk 634050, Russia
5	Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia

Funding (1)

Министерство науки и высшего образования РФ

075-02-2026-1339

In this paper, we introduce a two-variable parity polynomial invariant for virtual knotoids, defined on oriented virtual knotoid diagrams. The construction is based on the parity of classical crossings, where each crossing is classified as even or odd and treated accordingly in the definition of the invariant. We study several fundamental properties of this invariant. We demonstrate that the parity polynomial can distinguish pairs of virtual knotoids that are not distinguished by the odd writhe, prove that it is equivalent to the affine index polynomial, and establish that it is a Vassiliev invariant of order one. Finally, we give its relationship with the Petit gluing invariant.

Ding S. , Gao S. , Lei F. , Li F. , Vesnin A.
Two-Variable Parity Polynomial for Virtual Knotoids
Mathematics. 2026. V.14. N14. 2536 :1-26. DOI: 10.3390/math14142536 WOS OpenAlex

Submitted:	Jun 14, 2026
Accepted:	Jul 12, 2026
Published online:	Jul 14, 2026

≡ Web of science:	WOS:001832824100001
≡ OpenAlex:	W7168286002