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arXiv · 1802.07771

The Lattice of subracks is atomic

Amir Saki·Dariush Kiani
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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.

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Amir Saki, Dariush Kiani. 2018-02-21. The Lattice of subracks is atomic. https://arxiv.org/abs/1802.07771

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