The Lattice of subracks is atomic
Combinatorics
2018-02-23 v1 Commutative Algebra
Abstract
A rack is a set together with a self-distributive bijective binary operation. In this paper, we give a positive answer to a question due to Heckenberger, Shareshian and Welker. Indeed, we prove that the lattice of subracks of a rack is atomic. Further, by using the atoms, we associate certain quandles to racks. We also show that the lattice of subracks of a rack is isomorphic to the lattice of subracks of a quandle. Moreover, we show that the lattice of subracks of a rack is distributive if and only if its corresponding quandle is trivial. Finally, applying our corresponding quandles, we provide a coloring of certain knot diagrams.
Cite
@article{arxiv.1802.07771,
title = {The Lattice of subracks is atomic},
author = {Amir Saki and Dariush Kiani},
journal= {arXiv preprint arXiv:1802.07771},
year = {2018}
}
Comments
15 pages, 10 figures