TY - JOUR

T1 - Virtually symmetric representations and marked Gauss diagrams

AU - Bardakov, Valeriy G.

AU - Neshchadim, Mikhail V.

AU - Singh, Manpreet

N1 - Funding Information: Valeriy G. Bardakov is supported by Ministry of Science and Higher Education of Russia (agreement No. 075-02-2021-1392) and the Russian Science Foundation grant 19-41-02005. Mikhail V. Neshchadim is supported by the Russian Science Foundation grant 19-41-02005. Manpreet Singh was supported by IISER Mohali for the PhD Research fellowship. Manpreet Singh also thanks to his supervisor Dr. Mahender Singh for giving him the opportunity to attend VI Russian-Chinese Conference on Knot Theory and Related Topics at NSU (Novosibirsk) and 2nd International Conference on Groups and Quandles in low-dimensional topology at TSU (Tomsk) using his grant, where he had discussions with the first two authors. His visit to Russia was supported by the DST grant INT/RUS/RSF/P-02. Funding Information: Valeriy G. Bardakov is supported by Ministry of Science and Higher Education of Russia (agreement No. 075-02-2021-1392 ) and the Russian Science Foundation grant 19-41-02005 . Mikhail V. Neshchadim is supported by the Russian Science Foundation grant 19-41-02005 . Manpreet Singh was supported by IISER Mohali for the PhD Research fellowship. Manpreet Singh also thanks to his supervisor Dr. Mahender Singh for giving him the opportunity to attend VI Russian-Chinese Conference on Knot Theory and Related Topics at NSU (Novosibirsk) and 2nd International Conference on Groups and Quandles in low-dimensional topology at TSU (Tomsk) using his grant, where he had discussions with the first two authors. His visit to Russia was supported by the DST grant INT/RUS/RSF/P-02 . Publisher Copyright: © 2021 Elsevier B.V.

PY - 2022/2/1

Y1 - 2022/2/1

N2 - In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define the notion of virtual link groups which is an extension of virtual link groups defined by Kauffman. Moreover, we introduce the concept of marked Gauss diagrams as a generalisation of Gauss diagrams and their interpretation in terms of knot-like diagrams. We extend the definition of virtual link groups to marked Gauss diagrams and define their peripheral structure. We define Cm-groups and prove that every group presented by a 1-irreducible C1-presentation of deficiency 1 or 2 can be realised as the group of a marked Gauss diagram.

AB - In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define the notion of virtual link groups which is an extension of virtual link groups defined by Kauffman. Moreover, we introduce the concept of marked Gauss diagrams as a generalisation of Gauss diagrams and their interpretation in terms of knot-like diagrams. We extend the definition of virtual link groups to marked Gauss diagrams and define their peripheral structure. We define Cm-groups and prove that every group presented by a 1-irreducible C1-presentation of deficiency 1 or 2 can be realised as the group of a marked Gauss diagram.

KW - Marked Gauss diagram

KW - Marked link diagram

KW - Peripheral subgroup

KW - Virtual knot

KW - Virtual knot group

KW - Virtual spatial graph diagram

UR - http://www.scopus.com/inward/record.url?scp=85120830797&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2021.107936

DO - 10.1016/j.topol.2021.107936

M3 - Article

AN - SCOPUS:85120830797

VL - 306

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

M1 - 107936

ER -