Abstract: Twin groups are planar analogues of Artin braid groups and play a crucial role in the Alexander-Markov correspondence for the isotopy classes of immersed circles on the 2-sphere without triple and higher intersections. These groups admit diagrammatic representations, leading to maps obtained by the addition and deletion of strands. This paper explores Brunnian twin groups, which are subgroups of twin groups composed of twins that become trivial when any of their strands are deleted. We establish that Brunnian twin groups consisting of more than two strands are free groups. Furthermore, we provide a necessary and sufficient condition for a Brunnian doodle on the 2-sphere to be the closure of a Brunnian twin. Additionally, we delve into two generalizations of Brunnian twins, namely, κ-decomposable twins and Cohen twins, and prove some structural results about these groups. We also investigate a simplicial structure on pure twin groups that admits a simplicial homomorphism from Milnor’s construction of the simplicial 2-sphere. This gives a possibility to provide a combinatorial description of homotopy groups of the 2-sphere in terms of pure twins. © 2024, University at Albany. All rights reserved.

Cite: Bardakov V.G. , Kumar P. , Singh M.
Brunnian planar braids and simplicial groups
New York Journal of Mathematics. 2024. V.30. P.1235-1263. WOS Scopus

Dates:

Submitted:	Feb 12, 2024
Published print:	Sep 16, 2024
Published online:	Sep 16, 2024

Identifiers:

Web of science:	WOS:001317203900001
Scopus:	2-s2.0-85205784136

Citing: Пока нет цитирований