English

Plat closures of spherical braids in $\mathbb{R}P^3$

Geometric Topology 2023-11-15 v3

Abstract

We define plat closure for spherical braids to obtain links in RP3\mathbb{R}P^3 and prove that all links in RP3\mathbb{R}P^3 can be realized in this manner. Given a spherical braid β\beta of 2n2n strands in RP3\mathbb{R}P^3 we associate a permutation hβh_{\beta} on nn elements called \textit{residual permutation}. We prove that the number of components of the plat closure link of a spherical braid β\beta is same as the number of disjoint cycles in hβh_{\beta}. We also present a set of moves on spherical braids in the same spirit as the classical Markov moves on braids. The completeness of this set of moves to capture the entire isotopy classes of the plat closure links is still to be explored.

Keywords

Cite

@article{arxiv.2304.08954,
  title  = {Plat closures of spherical braids in $\mathbb{R}P^3$},
  author = {Rama Mishra and Visakh Narayanan},
  journal= {arXiv preprint arXiv:2304.08954},
  year   = {2023}
}

Comments

20 Pages, 22 Figures, preliminary version