The Lattice of subracks is atomic

TL;DR

This paper proves that the lattice of subracks of any rack is atomic, establishes an isomorphism with the lattice of subracks of associated quandles, and explores implications for knot diagram colorings.

Contribution

It demonstrates that the lattice of subracks is atomic and shows the isomorphism with subrack lattices of associated quandles, answering a question by Heckenberger, Shareshian, and Welker.

Findings

The lattice of subracks of a rack is atomic.

The lattice of subracks of a rack is isomorphic to that of an associated quandle.

The lattice is distributive if and only if the quandle is trivial.

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.