Tackling the Trefoils
Geometric Topology
2011-10-05 v1 Rings and Algebras
Abstract
The classical trefoil is famous for having a three-colouring which distinguishes it from the unknot. The three-colouring is also notorious for not distinguishing the right handed from the left handed trefoil. However with a bit of tweaking the three colours can also be used for this task. What lies behind the method is a new operation on biracks called doubling which converts the 3-colour quandle into a biquandle. Colouring with this biquandle distinguishes the right handed from the left handed trefoil. Equivalently it defines an element of the homology of the quandle or biquandle classifying space.
Cite
@article{arxiv.1110.0582,
title = {Tackling the Trefoils},
author = {Roger Fenn},
journal= {arXiv preprint arXiv:1110.0582},
year = {2011}
}