(1, 2) and weak (1, 3) homotopies on knot projections
Geometric Topology
2020-04-15 v1
Abstract
In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3).
Cite
@article{arxiv.2004.06312,
title = {(1, 2) and weak (1, 3) homotopies on knot projections},
author = {Noboru Ito and Yusuke Takimura},
journal= {arXiv preprint arXiv:2004.06312},
year = {2020}
}
Comments
13 pages, 25 figures